Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

What is the difference between the asymptotic behavior of an autonomous and a non-autonomous equation? This may be, at first glance, a simple question to answer. Let us discuss this problem a little. Consider a general non-autonomous differential equation given by

${\begin{matrix} \dot{u} = F (t, u), t ⩾ s, \\ u (s) = u_{0} \in X, \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {\dot u = F(t,u),\,t \geqslant s,}\\ {u(s) = {u_0} \in X,} \end{array}\right. \end{array}$$(1)

where X is a Banach space and F is in some metric space ℭ of functions. Assume also that for each u₀ ∈ X, s ∈ ℝ and F ∈ ℭ, the problem (1) has a uniquely defined solution u(t, s, F,u₀) for all times t ⩾ s and the map (t, s, u₀) ↦ u(t, s, F, u₀) is continuous for each F ∈ ℭ.

We know that, when F is independent of time, u(t, s, F, u₀) = u(t — s, 0, F, u₀), that is, the dependence of t and s in u are artificial, and in fact u depends only on the elapsed time t — s. Hence, the asymptotic behavior (the behavior for large times) can be obtained making t — ∞ or s → — ∞, indistinctly.

However, if F is time dependent, the dependence of t and s of the solution is explicit and the scenarios arising from making t → ∞ and s → — ∞ may be completely different. This is not so surprising once we realize that in the autonomous case (F independent of time) we have only one vector field, namely F, driving the solutions but, in the non-autonomous case we have infinitely many vector fields (F(t, ·) for each t) driving the solutions, and their behavior may be completely different for t → ∞ and s → —∞. This clarifies a little the understanding of asymptotic behavior for non-autonomous equations, and shows that it is not an easy task to study this subject. First, one must define which behavior will be treated: the forward attract/on (when t → ∞) or the pullback attract/on (when s → — ∞). Once the framework is set, we reach another problem: in any of them it is clear that the asymptotic behavior of the solutions of (1) are related with the behavior of the vector fields F(t, ·) when t → ±∞, which can be unrelated with the behavior of each F(t, ·). This could have been said in another way: the translates θ_sF alone are not enough, in general, to describe the asymptotic dynamics of (1) (for more details on this subject we refer to [7] or [9, 16, 17]). Thus, the next question comes quite naturally: how do we introduce the limiting vector fields of F(t, ·) in the study?

Consider ℭ the space of all functions H: ℝ × X → X, that are bounded in sets of the form ℝ × B, where B is a bounded set of X. Consider also the shift operatorθ_t: ℭ → ℭ given by

$θ_{t} H (\cdot, \cdot) = H (t + \cdot, \cdot), for each t \in ℝ .$ $$\begin{array}{} \displaystyle \theta_tH(\cdot,\cdot)=H(t+\cdot,\cdot), \hbox{ for each }t\in \mathbb{R}. \end{array}$$

Now define Σ₀ = {θ_tF}_t∈ℝ, which is the set of all translations of F and let

$Σ = closure of Σ_{0} in ℭ,$ $$\begin{array}{} \displaystyle \Sigma=\hbox{ closure of } \Sigma_0 \hbox{ in } \mathfrak{C}, \end{array}$$

which is known as the hull of F.

Using our assumptions for the problem (1), we know that each problem

${\begin{matrix} \dot{u} = H (t, u), for t \in ℝ \\ u (0) = u_{0} \in X, \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {\dot u = H(t,u),{\rm{\;for\;}}t \in \mathbb{R}}\\ {u(0) = {u_0} \in X,} \end{array}\right. \end{array}$$(2)

has a uniquely defined solution φ(t, H)u₀ for each t ⩾ 0, u₀ ∈ X and H ∈ Σ.

Note that we are now dealing with all the solutions of the problem (1) but also with all the solutions of the limiting vector fields of F(t, ·).

Remark 1.

To obtain problem (1), just consider H = θ_sF. The solutions ξ of (1) and ψ of (2) in this case are related by

$ψ (\cdot) = ξ (\cdot + s) .$ $$\begin{array}{} \displaystyle \psi ( \cdot ) = \xi ( \cdot + s). \end{array}$$

These objects, namely the solutions φ(t, H)u₀ in X and the shift operator θ_t in Σ

Clearly we can consider the restriction of the shift operator θ_t to Σ.

, give rise to what we call a non-autonomous dynamical system

See Definition 5.

. Several authors have studied this object, in the pursuit of fully understanding of the asymptotic dynamics of equation (1), and there are two distinct branches: the pullback approach (see, for example [16,22]), which deals with pullback attraction, and the uniform approacch (see, for example [28]). Each group has achieved several interesting results concerning asymptotic behavior of non-autonomous equations, and up until recently, these approaches seemed unrelated. In [5,7] the authors unify these results, presenting relations between these frameworks. To this end and to reach the full extent of this theory, they transform the non-autonomous dynamical system defined by φ and θ_t in an autonomous one, non-trivially, by defining

$Π (t) (u_{0}, H) = (φ (t, H) u_{0}, θ_{t} H),$ $$\begin{array}{} \displaystyle \Pi (t)({u_0},H) = (\varphi (t,H){u_0},{\theta _t}H), \end{array}$$(3)

which is called the skew-product semiflow, and study the autonomous semiflow defined by Π to obtain results for φ simply analysing the canonical projection in the first coordinate.

To be a little more precise, inside the study of non-autonomous equations such as (1) we can distinct at least four different notions of attractors, namely:

(i) the global attractor for the skew-product semiflow;

(ii) the pullback attractor for the evolution process.

(iii) the cocyle attractor for the non-autonomous dynamical system and

(iv) the uniform attractor for the non-autonomous system.

We will give a detailed description of each one of these objects in Section 2, as well as the relationships between these concepts, as done in [7], to describe the non-autonomous problems (1) in a very complete way.

1.2

Small perturbations

Imagine now that we have not only a single F(t, ·) but a family {H_ε(t, ·)}_ε∈[0,1] such that H_ε is close (in some sense) to F as ε → 0. Are we able to obtain results on the asymptotic behavior of the problems

${\begin{matrix} \dot{u} = H_{ϵ} (t, u), for t > 0 \\ u (s) = u_{0} \in X, \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {\dot u = {H_\varepsilon }\left( {t,u} \right),{\rm{ for }}t \gt 0} \hfill \\ {u\left( S \right) = {u_0} \in X,} \hfill \\ \end{array}\right. \end{array}$$(4)

for ε sufficiently small, given that we know the behavior of (1)?

Note that to study each problem one must perform all the previous discussion; that is, each ε will generate a different non-autonomous dynamical system and a skew-product semiflow. The question is: can we obtain results of continuity of the different types of asymptotic behavior as ε → 0?

This question, theoretical as is sounds, has a meaning in applications. Models in the real world are always approximations, due to data collection, empirical laws and simplifications, and thus, it is crucial that we are able to transfer properties from an equation to some small perturbations. Without this property, we have no guarantee whatsoever that the real phenomena will have a behavior close to our model.

In [5,7], the authors provide and extensive study on this topic, giving a detailed study of non-autonomous dynamical systems, different scenarios of asymptotic behavior and relationships among them, extracting informations from the skew-product semiflow and transporting them to the non-autonomous dynamical system. Also, the reader can find a deep study of continuity of small perturbations of non-autonomous system, but arising from autonomous equations (see for instance [7, 15, 16]).

The study of non-autonomous perturbations of non-autonomous dynamical systems directly is still an almost blank page, and in this paper we give some steps in this direction, by studying non-autonomous perturbations of a non-autonomous equation, to provide results of continuity of the asymptotic behavior using the framework discussed above.

It is not known, so far, how to do the general theory when we consider non-autonomous perturbations of a non-autonomous system. In examples, we can see however that few steps are clear: first one must be able to prove the global existence and uniqueness of solutions not only for the equation in question, but also for all the limiting vector fields associated with the non-linearity - this step is the key for the development of the following results - to be able to construct a non-autonomous system. Then we must be able, with some uniformity on the vector fields, to obtain an uniform estimate that allow us to find a compact set that attracts all the solutions, independently of the vector field. Once this is done, we can find such an attractor for the associated skew-product semiflow and with this object at hand, we can try to understand all the asymptotic behaviors of our equation. Dealing with perturbations adds a difficulty to this process, since we must be able to do such study for each ε small and obtain the result with uniformity in this parameter.

1.3

Non-autonomous non-classical parabolic equations

As mentioned before, a general theory for the study of non-autonomous perturbations of non-autonomous models is not available at this point, so we will put our best efforts to understand in some elaborated examples, in order to obtain a deep understanding of such perturbations. The problem we will deal with in this paper is to study non-autonomous perturbations of some non-autonomous parabolic equations.

Non-classical parabolic equations arise as models describing physical phenomena such as non-Newtonian flow, soil mechanics, heat conduction, etc. (see, for instance, [1—1,8,21,23,25,29,30] and references therein). We will focus our study in non-autonomous perturbations of the following non-classical non-autonomous parabolic equation

${\begin{matrix} u_{t} - γ (t) Δ u_{t} - Δ u = f (u), in Ω \\ u = 0, on \partial Ω \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma (t)\Delta {u_t} - \Delta u = f(u),{\rm{\;in\;}}\Omega }\\ {u = 0,{\rm{\;on\;}}\partial \Omega } \end{array}\right. \end{array}$$(5)

where Ω ⊂ ℝⁿ is a smooth bounded domain, for some n ⩾ 3, with f and γ satisfying some suitable conditions. More specifically, we will deal with perturbations of the form

${\begin{matrix} u_{t} - γ (t) Δ u_{t} - Δ u = g_{ϵ} (t, u), in Ω \\ u = 0, on \partial Ω \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega } \hfill \\ \end{array}\right. \end{array}$$(6)

where {g_ε}_{ε∈[0, 1]} is a family of non-autonomous functions satisfying some continuity conditions.

In the work of Aifantis et al., [1-3] we can find a quite general approach to deduce these equations in the autonomous case without delay. In the aforementioned papers, it is pointed out that the classical reaction-diffusion equation

$u_{t} - Δ u = g (u)$ $$\begin{array}{} \displaystyle {u_t} - \Delta u = g(u) \end{array}$$

does not contain each aspect of the reaction-diffusion problem, and it neglects viscidity, elasticity, and pressure of medium in the process of solid diffusion. The authors obtained a diffusion theory similar to Fick’s classical model for solute in an undisturbed solid matrix, obtaining a hyperbolic equation

$u_{t} + D_{1} u_{t t} = D_{2} Δ u,$ $$\begin{array}{} \displaystyle {u_t} + {D_1}{u_{tt}} = {D_2}\Delta u, \end{array}$$

where D₁ and D₂ are positive constants. Assign viscosity to the diffusing substance, they arrived to que following equation

$u_{t} + D_{1} u_{t t} = D_{2} Δ u + D_{3} Δ u_{t},$ $$\begin{array}{} \displaystyle {u_t} + {D_1}{u_{tt}} = {D_2}\Delta u + {D_3}\Delta {u_t}, \end{array}$$

and neglecting the inertia term, finally obtained the non-classical parabolic equation

$u_{t} = D_{2} Δ u + D_{3} Δ u_{t},$ $$\begin{array}{} \displaystyle {u_t} = {D_2}\Delta u + {D_3}\Delta {u_t}, \end{array}$$

where D₃ is also a positive constant.

The asymptotic behavior of the model without delay terms and with constant coefficients

$u_{t} - μ Δ u_{t} - Δ u + g (u) = f (x), μ \in [0, 1]$ $$\begin{array}{} \displaystyle {u_t} - \mu \Delta {u_t} - \Delta u + g(u) = f(x),\qquad \mu \in [0,1] \end{array}$$

is studied in [31], where, in particular, it is shown the well-posedness of the problem and the existence of the global attractor either in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$ or in H²(Ω), depending on the regularity of the initial data. They also showed the continuity of the global attractor in Hausdorff semidistance when μ → 0 in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$).

The introduction of a time dependence in coefficient γ(t) represents the variability of viscosity in time due to, for example, external environmental temperatures. This time dependence provides the system with a non-autonomous nature.

The study of a non-autonomous case with delay appeared in [12] for the first time, where it was established the well-posedness of the problem when γ(t) ≡ γ is constant.

In [26], Rivero studied the existence of the pullback attractor and its continuity under non-autonomous perturbations, showing the existence of a concrete structure under some assumptions on the non-linearity and giving a first approach to the study of perturbations in non-autonomous problems.

Remark 2.

This example is understood by us as a good starting point to the study of non-autonomous perturbations. Mainly because (2) is a non-autonomous equation, but term that causes this phenomena (the function γ) has no effect on the equilibria, which are the equilibria of the elliptic equation - Δu = f(u).

1.4

Novelties

In this paper we give a step towards understanding non-autonomous perturbations of non-autonomous equations. We first study the problem (6) for each ε ∈ [0, 1] using the ideas presented in Subsection 1.1, but always having the discussion of Subsection 1.2 in mind, that is, not only we will deal with each equation separately for each ε, but also we have to take into account that we must be able to obtain the results with uniformity for ε ∈ [0, 1]. Using this, we will be able to obtain continuity results for the family of equations given by (6). In the next section, we will present a detailed description of our main results.

Remark 3.

One important thing to stress out is that, even that f does not depend on t, the function γ makes problem (5) non-autonomous.

1.5

Description of the main results

To describe the contents of our work and to state the main results, we first make some assumptions on the functions γ, f and g_ε as follows: suppose that γ: ℝ → (0, ∞) is a uniformly continuous function which satisfies 0 < γ₀ ⩽ γ(t) ⩽ γ₁ < ∞ and the family {g_ε}_{εε[0, 1]} of continuously differentiable functions from ℝ² to ℝ with g₀(t, s) = f (s) for all t, s ε ℝ, that satisfies

$| g_{ϵ} (t, s_{1}) - g_{ϵ} (t, s_{2}) | ⩽ α | s_{1} - s_{2} | (1 + | s_{1} |^{ρ - 1} + | s_{2} |^{ρ - 1}),$ $$\begin{array}{} \displaystyle |g_\epsilon(t,s_1)-g_\epsilon(t,s_2)|\leqslant \alpha|s_1-s_2|(1+|s_1|^{\rho-1}+|s_2|^{\rho-1}), \end{array}$$(H1)

$\underset{| s | \to \infty}{\lim \sup} \frac{g_{ϵ} (t, s)}{s} ⩽ δ < λ_{1},$ $$\begin{array}{} \displaystyle \limsup_{|s|\to \infty} \frac{g_\epsilon(t,s)}{s} \leqslant \delta < \lambda_1, \end{array}$$(H2)

$\int_{0}^{\infty} \frac{\partial}{\partial t} g_{ϵ} (t, s) d s < \infty$ $$\begin{array}{} \displaystyle \int_{0}^\infty \frac{\partial}{\partial t} g_\epsilon(t,s)ds<\infty \end{array}$$(H3)

Also we assume that there exists a bounded function β defined in the interval [0, 1] with β (δ) → 0, as δ → 0⁺, and satisfying

$\sup_{t \in ℝ} | g_{ϵ} (t, s) - f (s) | ⩽ β (ϵ) (1 + | s |^{ρ - 1}), for all s \in ℝ and ϵ \in [0, 1],$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|g_\epsilon(t,s)-f(s)|\leqslant \beta(\epsilon)(1+|s|^{\rho-1}), \hbox{ for all }s\in \mathbb{R} \hbox{ and } \epsilon\in [0,1], \end{array}$$(H4)

and

$\sup_{t \in ℝ} | \partial_{s} g_{ϵ} (t, s) - f^{'} (s) | ⩽ β (ϵ) (1 + | s |^{ρ - 1}), for all s \in ℝ and ϵ \in [0, 1],$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|\partial_sg_\epsilon(t,s)-f'(s)|\leqslant \beta(\epsilon)(1+|s|^{\rho-1}), \hbox{ for all }s\in \mathbb{R} \hbox{ and } \epsilon\in [0,1], \end{array}$$(H5)

where λ₁ > 0 is the first eigenvalue of the negative Laplacian A = —Δ with Dirichlet boundary condition, for some α > 0 and $1 ⩽ ρ < \frac{n + 2}{n - 2}$ $\begin{array}{} \displaystyle 1\leqslant \rho< \frac{n+2}{n-2} \end{array}$, with (H1), (H2) uniformly for t ∈ ℝ and ε ∈ [0, 1] and (H3) uniformly for ε ∈ [0, 1].

In Section 2 we present a brief summary about the theory of autonomous and non-autonomous systems, with their respective attractors. In Section 3 we will describe the precise spaces, along with the required topologies, to fit the family of non-autonomous non-classical equations in the framework described in Subsection 1.1. Sections 4 and 5 are devoted to prove that each equation (6) generates a non-autonomous dynamical system and prove the existence of the several types of attractors described in Section 2, respectively.

In Sections 6 and 7, motivated by the discussion in Subsection 1.2, we study the upper semicontinuity and topological structural stability of each kind of global attractor found in Section 5, respectively. With all this work, we provide a complete study for the various scenarios of asymptotic behavior for non-autonomous dynamical systems described in Subsection 1.1.

Preliminaries: Asymptotic dynamics of non-autonomous equations

We will briefly present the theory described in [7], which studies non-autonomous differential equations in different frameworks and gives relations between these dynamics.

2.1

Semigroups

First of all, we define the notions of semigroups and their global attractors (the reader may see [18] for more details of this theory).

Let (X, d) be a metric space and 𝒞(X) the set of all continuous maps from X into itself. A semigroup in X is a one parameter family {T(t): t ⩾ 0} such that

(a)T (0) = Id_X, with I_X being the identity in X,

(b)T(t)T(s) = T(t + s), for all t, s ⩾ 0 and

From now on we are going to denote by d_H(·, ·) the Hausdorff semidistance between two subsets of X, that is, for any A, B ⊂ X:

$d_{H} (A, B) = \sup_{a \in A} \inf_{b \in B} d (a, b) .$ $$\begin{array}{} \displaystyle {\rm d}_H(A,B)=\displaystyle{\sup_{a\in A}\inf_{b\in B}} \ d(a,b). \end{array}$$

Definition 1.

A compact set 𝒜 is called a global attractor of {T(t): t ⩾ 0} if satisfies:

(i)𝒜 is invariant for {T(t): t ⩾ 0}; that is, T(t)𝒜 = 𝒜, for all t ⩾ 0.

(ii)𝒜attracts bounded subsets under the action of {T(t): t ⩾ 0}; that is, for each bounded subset B of X we have

$\lim_{t \to \infty} d_{H} (T (t) B, A) = 0.$ $$\begin{array}{} \displaystyle \lim _{t\to \infty } {\rm d}_H(T(t)B, \mathscr{A})=0. \end{array}$$

The global attractor of a semigroup describes the asymptotic behavior of the semigroup. To be more precise, we define a global solution of {T(t): t ⩾ 0} as a function ξ : ℝ → X such that T(t)ξ (s) = ξ (t + s) for all t ⩾ 0 and s ∈ ℝ. Then, we know that if 𝒜 is the global attractor of {T(t): t ⩾ 0} we have

$A = {x \in X : x = ξ (0) for some bounded global solution ξ of {T (t) : t ⩾ 0}} .$ $$\begin{array}{} \displaystyle {\scr A} = \{ x \in X:x = \xi (0){\;\text{for some bounded global solution }}\; \xi \;{\text{of}}\; \{ T(t):t \geqslant 0\} \} . \end{array}$$

This characterization means that not only the global attractor attracts all positive orbits {T(t)x: t ⩾ 0} (x ∈ X) but it actually consists of all bounded globally defined solutions. Moreover, the global attractor for a semigroup is unique.

To obtain existence of global attractors for semigroups, we will need some definitions.

Definition 2.

Let B, C ⊂ X. We say that BabsorbsC under the action of {T(t):t ⩾ 0} if there exists T ⩾ 0 such that

$T (t) C \subset B, for all t ⩾ T .$ $$\begin{array}{} \displaystyle T(t)C\subset B, \hbox{ for all } t\geqslant T. \end{array}$$

Definition 3.

We say that a semigroup {T(t): t ⩾ 0} is asymptotically compact if given sequences t_n → ∞ and {x_n}_n∈𝔹 bounded in X such that {T(t_n)x_n}_n∈ℕ is bounded, then {T(t_n)x_n}_n∈ℕ is precompact in X.

With these definitions we are able to state the main result about existence of global attractors, that we be needed later.

Theorem 1.

(Theorem 3.4 in [18]). Let {T(t): t ⩾ 0} be an asymptotically compact semigroup. Assume that there exists a bounded set B ⊂ X such that B absorbs all bounded subsets of X under the action of {T(t): t ⩾ 0}. Then {T(t): t ⩾ 0} has a global attractor 𝒜 and 𝒜 ⊂ B.

2.2

Evolutions processes

Now we are going to define evolution processes and their pullback atractors (see [9,16] for more details). These concepts appear in the literature as natural generalizations for semigroups and global attractors, respectively.

Again, let (X, d) be a metric space. An evolution process in X is a two parameter family {T(t, s): t ⩾ s} in 𝒞(X) such that

(a)T(t, t) = Id_X,

(b)T(t, s)T(s, τ) = T(t, τ), for all t ⩾ s ⩾ τ and

Definition 4.

A family of compact sets {A(t)}_t∈ℝ is called a pullback attractor of T(t,s) if satisfies:

(i) {A(t)}_t∈ℝ is invariant; that is, T(t,s)A(s) = A(t), for all t ⩾ s.

(ii) {A(t)}_t∈ℝpullback attracts bounded subsets; that is, for each bounded subset B of X and t ∈ ℝ, we have

$\lim_{s \to - \infty} d_{H} (T (t, s) B, A (t)) = 0.$ $$\begin{array}{} \displaystyle \mathop {\lim }\limits_{s \to - \infty } {{\rm{d}}_H}(T(t,s)B,A\left( t \right)) = 0. \end{array}$$

(iii) {A(t)}_t∈ℝ is the minimal family of closed sets with property (ii).

Remark 4.

We note that when T(t,s) = S(t − s), the family {S(t): t ⩾ 0} is a semigroup in X and Definition 4 reduces to the definition of global attractors. The first difference that appears is item (iii) in Definition 4 and it ensures the uniqueness of the pullback attractor, since it does not follows directly from (i) and (ii) as in the autonomous case.

2.3

Non-autonomous dynamical systems

Now, we will introduce the concept of non- autonomous dynamical systems, which is a general method that provides a way to form the base space for a given non-autonomous differential equation. The idea of this method is to consider the family of non-linearities as a base flow driven by the time shift.

Definition 5.

A non-autonomous dynamical system (NDS) is a quadruple (φ, θ)_(X,Σ), where X, Σ are a metric spaces with metrics d_X and d_Σ, respectively; θ ≐ (θ_t: t ⩾ 0} is a semigroup in Σ, called the shift operator (or driving semigroup), and φ: ℝ⁺ × Σ × X → X is a map

Note that we use the notation φ(t, σ, x) = φ(t, σ)x for all (t, σ, x) ∈ ℝ⁺ × Σ × X.

that verifies

(i) φ (0, σ) = Id_X for all σ ∈ Σ;

(ii) ℝ⁺ × Σ ∋ (t, σ) ↦ φ(t, σ)u ∈ X is continuous, and

(iii) φ(t + s, σ) = φ(t, θ, σ)φ(s, σ), for all t, s ⩾ 0 and σ ∈ Σ.

The map φ is called the cocycle semiflow and property (iii) is know as the cocycle property.

To define the cocycle attractor and the uniform attractor for a NDS (φ, θ)_(X,Σ), we first must define the concepts of non-autonomous set, invariance and pullback attraction in this framework:

Definition 6.

A non-autonomous set is a family {D(σ)}_σ∈Σ of subsets of X indexed in Σ. We say that (D(σ)}_σ∈Σ is an open (closed, compact) non-autonomous set if each fiberD(σ) is an open (closed, compact) subset of X.

Definition 7.

A non-autonomous set {D(σ)}_σ∈Σ is invariant under the NDS (φ, θ)_(X,Σ) if

$φ (t, σ) D (σ) = D (θ_{t} σ), for all t ⩾ 0 and σ \in Σ .$ $$\begin{array}{} \displaystyle \varphi (t,\sigma )D(\sigma ) = D({\theta _t}\sigma ), \;{\text{for all }}\;t \geqslant 0\;{\rm{and}}\; \sigma \in \Sigma . \end{array}$$

To define the concept of pullback attraction, we must ask some additional properties on Σ, and from now on, we are going to assume that Σ, is compact and invariant for the driving semigroup (θ:t ⩾ 0}, and also that (θ_t: t ⩾ 0} is a group over Σ; that is, θ_t is invertible, and we denote $θ_{t}^{- 1} = θ_{- t}$ $\begin{array}{} \displaystyle \theta _t^{ - 1} = {\theta _{ - t}} \end{array}$.

Remark 5.

Actually, these assumptions can be dropped. We can obtain the same results requiring only that {θ_t : t ⩾ 0} possess a global attractor in Σ, with virtually no additional work, but with a more difficult notation. So, for simplicity, we shall assume all the hypotheses above.

Definition 8.

A compact non-autonomous set {A(σ)}_σ∈Σ is called a cocycle attractor of (φ, θ)_(X,Σ) if

(i) {A(σ)}_σ∈Σ is invariant under (φ, θ)_(X,Σ);

(ii) {A(σ)}_σ∈Σ pullback attracts all bounded subsets B ⊂ X, i.e.

$\lim_{t \to + \infty} d_{H} (φ (t, θ_{- t} σ) B, A (σ)) = 0.$ $$\begin{array}{} \displaystyle \lim _{t\to +\infty }{\rm d }_H (\varphi (t,\theta _{-t}\sigma)B,A(\sigma))=0. \end{array}$$

(iii) {A(σ)}_σ∈Σ is the minimal among the closed non-autonomous sets with property (ii).

We can also deal with the uniform attraction for a NDS - in this framework, the attraction do not depend on the chosen σ ∈ Σ, - that is, we say that the subset K ⊆ X is uniform attracting for the NDS (φ, σ)_(X,Σ) if for each B ⊂ X bounded,

$\lim_{t \to \infty} \sup_{σ \in Σ} d_{H} (φ (t, σ) B, K) = 0.$ $$\begin{array}{} \displaystyle \lim_{t\to\infty}\sup_{\sigma\in\Sigma}{\rm d}_H(\varphi(t,\sigma)B,K) = 0. \end{array}$$

Definition 9.

A compact subset 𝒜 ⊂ X is called the uniform attractor for the NDS (φ, σ)_(X,Σ) if it is the minimal closed subset of X that uniform attracts all bounded subsets of X.

Theorem 2.

A NDS (φ, σ)_(XΣ)has a uniform attractor if and only if there exists a compact uniform attracting set K.

2.4

Skew-product semiflows

When dealing with non-autonomous dynamical systems, it is worthwhile to question if we can transform them into an autonomous one; and that is precisely the case: for a given NDS (φ, θ)_(X,Σ) we can define a semigroup {Π(t): t ⩾ 0} in the product space $X = X \times Σ$ $\begin{array}{} \displaystyle \mathbb X=X\times\Sigma \end{array}$ as (see [27,28] for more details)

$Π (t) (u, σ) = (φ (t, σ) u, θ_{t} σ),$ $$\begin{array}{} \displaystyle \Pi(t) (u,\sigma) = (\varphi(t,\sigma)u,\theta_t\sigma), \end{array}$$(7)

which is called skew-product semiflow associated with (φ, θ)_(X,Σ).

So far we obtained three different objects:

the evolution process {T(t, s): t ⩾ s};

the non-autonomous dynamical system (φ, θ)_(X,Σ) and

the skew-product semiflow {Π(t): t ⩾ 0},

and the four different notions of ‘attractors’ listed in the Introduction:

(i) the pullback attractor {A(t)}_t∈ℝ of {T(t, s): t ⩾ s};

(ii) the cocycle attractor {A(σ)}_σ∈Σ of (φ, θ)_(X,Σ);

(iii) the uniform attractor 𝒜 of (φ, θ)_(X,Σ) and

(iv) the global attractor 𝔸 of {Π(t): t ⩾ 0},

and now we presente briefly the relationships between these objects (as in [7]). We begin with the relation between the global attractor of {Π(t): t ⩾ 0} and the cocycle attractor of (φ, θ)_(X,Σ).

Theorem 3.

(Propositions 3.30 and 3.31 in [22], or Theorem 3.4 in [14]). Let (φ, θ)_(X,Σ)be a non-autonomous dynamical system and let {Π(t): t ⩾ 0} be the associated skew product semiflow on X × Σ with a global attractor 𝔸. Then {A(σ)}_σ∈Σwith A(σ) = {x ∈ X : (x, σ) ∈ 𝔸} is the cocycle attractor of (φ, θ)_(X,Σ).

The following theorem shows the relationship between the global attractor of a skew product semiflow and the pullback attractors of the evolution processes it may contain.

Theorem 4.

(Theorem 2.7 in [5]). Assume that the skew product semiflow {Π(t): t ⩾ 0} possesses a global attractor 𝔸. Then the evolution process {T_σ(t, s) : t ⩾ s} given by

$T_{σ} (t, s) u = φ (t - s, θ_{s} σ) u, u \in X,$ $$\begin{array}{} \displaystyle T_{\sigma}(t,s)u= \varphi(t-s,\theta_s\sigma)u, \ u\in X, \end{array}$$

possesses a pullback attractor {A_σ(t)}_t∈ℝ. Moreover

$A = \underset{σ \in Σ}{\cup} [\underset{t \in ℝ}{\cup} A_{σ} (t) \times {σ}] .$ $$\begin{array}{} \displaystyle \mathbb{A}=\bigcup_{\sigma\in\Sigma}\left[\bigcup_{t\in \mathbb{R}}{A}_\sigma(t)\times \{\sigma\}\right]. \end{array}$$

Therefore, if 𝔸 is the global attractor of the associated semigroup {Π(t) : t ∈ ℝ} of (φ, σ)_(X,Σ), then the uniform attractor 𝒜 is the projection on X of the global attractor, that is, 𝒜 = π_X(𝔸), where π_X : X × Σ → X is the projection over X.

Now, the relationship between the uniform attractor and the pullback attractor is clear.

Theorem 5.

The NDS (φ, σ)_(X,Σ)has a uniform attractor 𝒜 if and only if the associated skew-product semiflow {Π(t): t ⩾ 0} has a global attractor 𝔸 and

$A = π_{X} (A) = \underset{σ \in Σ}{\cup} \underset{t \in ℝ}{\cup} A_{σ} (t)$ $$\begin{array}{} \displaystyle \mathscr A = \pi_X(\mathbb A) = \bigcup_{\sigma\in\Sigma}\bigcup_{t\in \mathbb {R}}{A}_\sigma(t) \end{array}$$

The following result shows us the required assumptions in order to obtain the existence of the global attractoi for the skew-product semiflow {Π(t) : t ⩾ 0} associated with the NDS (φ, σ)_(X,Σ) based on the existence of it cocycle attractor (see [14,22] for more details).

Theorem 6.

Suppose that {A(σ)}_σ∈Σis the cocycle attractor of (φ, θ)_(X,Σ), {Π(t) : t ⩾ 0} is the associated skew-product semiflow. Assume that {A(σ)}_σ∈Σis uniformly attracting, i.e.,

$\lim_{t \to + \infty} \sup_{σ \in Σ} dist (φ (t, θ_{- t} σ) D, A (σ)) = 0,$ $$\begin{array}{} \displaystyle \mathop {\lim }\limits_{t \to + \infty } \mathop {\sup }\limits_{\sigma \in \Sigma } {\rm{dist}}(\varphi (t,{\theta _{ - t}}\sigma )D,A(\sigma )) = 0, \end{array}$$

and that ∪_σ∈ΣA(σ) is precompact in X. Then the set 𝔸 associated with {A(σ)}_σ∈Σ, given by

$A = \underset{σ \in Σ}{\cup} A (σ) \times {σ},$ $$\begin{array}{} \displaystyle \mathbb{A}=\bigcup_{\sigma\in \Sigma}A(\sigma)\times \{\sigma\}, \end{array}$$

is the global attractor of {Π(t) : t ⩾ 0}.

With these results we complete the relations between the four different asymptotic dynamics we presented. In this paper, as we said before, we will try to deal with these four dynamics, to obtain as much information as we can of equation (6).

Driving semigroups of translations for (6)

In this section we will put the family of equations (6) in the framework described on Subsection 1.1. To this end, let A = −Δ : D(A) ⊂ X → X be the negative Laplacian operator with Dirichlet boundary condition, defined in $D (A) = H_{0}^{1} (Ω) \cap H^{2} (Ω)$ $\begin{array}{} \displaystyle D(A) = H_0^1(\Omega ) \cap {H^2}(\Omega ) \end{array}$, where X = L²(Ω). Consider the fractional power scale X^α, with α ∈ ℝ, generated by (X,A). Also, consider the Nemytskii operators $g_{ϵ}^{e} (\cdot, \cdot)$ $\begin{array}{} \displaystyle g^e_\epsilon(\cdot,\cdot) \end{array}$ defined as $g_{ϵ}^{e} (t, u) (x) = g_{ϵ} (t, u (x))$ $\begin{array}{} \displaystyle g^e_\epsilon(t,u)(x)=g_\epsilon(t,u(x)) \end{array}$ for each (t, x) ∈ ℝ × Ω and u : Ω → ℝ.

Following the ideas in [26], we define the operators

$B_{γ} (t) = {(I + γ (t) A)}^{- 1} and {\tilde{A}}_{γ} (t) = A B_{γ} (t),$ $$\begin{array}{} \displaystyle {B_\gamma }(t) = {(I + \gamma (t)A)^{ - 1}}{\rm{ and }}{\tilde A_\gamma }(t) = A{B_\gamma }(t), \end{array}$$

and the function $g_{ϵ, γ} (t, u) = B_{γ} (t) g_{ϵ}^{e} (t, u)$ $\begin{array}{} \displaystyle g_{\epsilon,\gamma}(t,u)=B_\gamma(t)g_{\epsilon}^e(t,u) \end{array}$, for each ε ∈ [0,1], we can write problem (6) as

$u_{t} = F_{ϵ} (t, u),$ $$\begin{array}{} \displaystyle u_t=F_\epsilon(t,u), \end{array}$$(8)

where F_ε(t, u) = −Ã_γ(t)u + g_ε,γ(t, u). The domain of the operators Ã_γ(t) does not depend on time and the operators ℝ ∋ t ↦ B_γ(t) and ℝ ∋ t ↦ Ã_γ(t) are absolutely continuous functions.

To study (6), for each ε ∈ [0,1], we must be able to study the hull of the function F_ε(·,·) in a suitable space. This is our task in the next subsection.

3.1

Driving groups of translations

We begin considering the sets

𝒞₁ = C_b(ℝ, ℝ) of continuous bounded functions from ℝ to itself with metric

$d_{1} (λ_{1}, λ_{2}) = \sup_{t \in ℝ} | λ_{1} (t) - λ_{2} (t) |;$ $$\begin{array}{} \displaystyle d_1(\lambda_1,\lambda_2)=\sup_{t\in \mathbb{R}}|\lambda_1(t)-\lambda_2(t)|; \end{array}$$

𝒞₂ of continuous functions from ℝ² to ℝ, which satisfies: h ∈ 𝒞₂ if there exists constants γ,ε ⩾ 0 such that

$\sup_{t \in ℝ} | h (t, s_{1}) - h (t, s_{2}) | ⩽ γ | s_{1} - s_{2} | (1 + | s_{1} |^{ρ - 1} + | s_{2} |^{ρ - 1}), for all s_{1}, s_{2} \in ℝ,$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|h(t,s_1)-h(t,s_2)|\leqslant \gamma|s_1-s_2|(1+|s_1|^{\rho-1}+|s_2|^{\rho-1}), \hbox{ for all }s_1,s_2\in \mathbb{R}, \end{array}$$

and

$\sup_{t \in ℝ} | h (t, 0) | ⩽ ε .$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|h(t,0)|\leqslant \omega. \end{array}$$

In 𝒞₂ we introduce the norm

$‖ h ‖_{C_{2}} = \sup_{\underset{s \neq 0}{t, s \in ℝ}} \frac{| h (t, s) - h (t, 0) |}{| s | (1 + | s |^{ρ - 1})} + \sup_{t \in ℝ} | h (t, 0) |,$ $$\begin{array}{} \displaystyle \|h\|_{\mathcal{C}_2}=\sup_{\underset{s\neq 0}{t,s\in \mathbb{R}}}\frac{|h(t,s)-h(t,0)|}{|s|(1+|s|^{\rho-1})}+\sup_{t\in \mathbb{R}}|h(t,0)|, \end{array}$$

and the distance

$d_{2} (h_{1}, h_{2}) = ‖ h_{1} - h_{2} ‖_{C_{2}} .$ $$\begin{array}{} \displaystyle {d_2}({h_1},{h_2}) = \|{h_1} - {h_2}\|{_{{{\cal C}_2}}}. \end{array}$$

Remark 6.

We have that 𝒞₁ and 𝒞₂ are Banach spaces with norms ||λ||_𝒞₁ = sup_t∊ℝ |λ(t)| and ||·||𝒞₂, respectively.

Clearly we have that γ ∊ 𝒞₁ and g_ε ∊ 𝒞₂, for all ε ∊ [0,1]. We define the group of translations in both

Here we denote both groups the same, since there will be no confusion of notation.

𝒞₁ and 𝒞₂, {θ_t: t ∊ ℝ}, by

$θ_{t} λ (s) = λ (t + s), for all λ \in C_{1} and t, s \in ℝ;$ $$\begin{array}{} \displaystyle \theta_t\lambda (s)=\lambda (t+s), \hbox{ for all }\lambda \in \mathcal{C}_1 \hbox{ and }t,s\in \mathbb{R}; \end{array}$$

and

$θ_{t} h (r, s) = h (t + r, s), for all h \in C_{2} and t, r, s \in ℝ .$ $$\begin{array}{} \displaystyle \theta_th(r,s)=h(t+r,s), \hbox{ for all } h\in \mathcal{C}_2 \hbox{ and } t,r,s\in \mathbb{R}. \end{array}$$

Also, let

Γ be the hull of γ in 𝒞₁; that is,

$Γ = closure of {θ_{t} γ}_{t \in ℝ} in (C_{1}, d_{1}) .$ $$\begin{array}{} \displaystyle \Gamma=\hbox{ closure of } \{\theta_t\gamma\}_{t\in \mathbb{R}} \hbox{ in } (\mathcal{C}_1,d_1). \end{array}$$

𝒢_ε be the hull of g_ε in 𝒞₂; that is,

$G_{ϵ} = closure of {θ_{t} g_{ϵ}}_{t \in ℝ} in (C_{2}, d_{2}) .$ $$\begin{array}{} \displaystyle \mathscr{G}_\epsilon= \hbox{ closure of } \{\theta_tg_\epsilon\}_{t\in \mathbb{R}} \hbox{ in } (\mathscr{C}_2,d_2). \end{array}$$

Remark 7.

Since γ is bounded and uniformly continuous on ℝ, the set Γ is compact in (𝒞₁, d₁), using the Arzelá-Ascoli Theorem.

Note that, by simple computations, we have that there exists a constant C > 0 such that

$\sup_{t \in ℝ} ‖ B_{λ_{1}} (t) - B_{λ_{2}} (t) ‖_{L (H^{- 1}, H_{0}^{1} (Ω))} ⩽ C ‖ λ_{1} - λ_{2} ‖_{C_{1}}$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\|B_{\lambda_1}(t)-B_{\lambda_2}(t)\|_{\mathscr{L}(H^{-1},H^1_0(\Omega))}\leqslant C\|\lambda_1-\lambda_2\|_{\mathscr{C}_1} \end{array}$$

and

$\sup_{t \in ℝ} ‖ {\tilde{A}}_{λ_{1}} (t) - {\tilde{A}}_{λ_{2}} (t) ‖_{L (H_{0}^{1} (Ω))} ⩽ C ‖ λ_{1} - λ_{2} ‖_{C_{1}},$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\|\tilde{A}_{\lambda_1}(t)-\tilde{A}_{\lambda_2}(t)\|_{\mathscr{L}(H^1_0(\Omega))}\leqslant C\|\lambda_1-\lambda_2\|_{\mathscr{C}_1}, \end{array}$$

for all λ₁, λ₂ ∊ Γ, where H^—1 is the dual space of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H^1_0(\Omega) \end{array}$.

Since g₀(t, s) = f(s) for all t, s ∊ ℝ, 𝒢₀ = {f} and hence it is compact in (𝒢₂, d₂).

Since hypotheses (H1)-(H4) are uniform for t ∊ ∨, they all are satisfied by every function in 𝒢_ε.

Using items 1 and 2 from the previous remark, we will make one additional assumption on the family {𝒢_ε}:

$G_{ϵ} is compact in (C_{2}, d_{2}) for each ϵ \in (0, 1] .$ $$\begin{array}{} \displaystyle \mathscr{G}_\epsilon \hbox{ is compact in } (\mathscr{C}_2,d_2) \hbox{ for each } \epsilon\in (0,1]. \end{array}$$(C)

Remark 8.

Condition (C) is verified, for instance, when each g_ε is time-independent, or periodic in t, or almost-periodic in t (for the latter, see for instance [20, Appendix - Theorem 11]).

Now we can see that each function h_ε ∊ 𝒢_ε defines a Nemytskii operator from $ℝ \times X^{\frac{s}{2}}$ $\begin{array}{} \displaystyle \mathbb{R}\times X^{\frac s2} \end{array}$ into $L^{\frac{2 n}{n + 2 r}} (Ω)$ $\begin{array}{} \displaystyle L^{\frac{2n}{n+2r}}(\Omega) \end{array}$, for suitable s and r.

Lemma 7.

Assume that the family {𝒢_ε}_ε∊[0,1]satisfies (H1), A is the negative Dirichlet Laplacian in X with domain $X^{1} = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle X^1=H^2(\Omega)\cap H^{1}_0(\Omega) \end{array}$and consider its closed extension to $H^{- r} = {(X^{\frac{r}{2}})}^{'}$ $\begin{array}{} \displaystyle H^{-r}=(X^{\frac{r}{2}})' \end{array}$, the dual space of $X^{\frac{r}{2}}$ $\begin{array}{} \displaystyle X^{\frac r2} \end{array}$, (in particular, $H^{- 1} = H_{0}^{1} (Ω)^{'})$ $\begin{array}{} \displaystyle H^{-1}=H^1_0(\Omega)') \end{array}$. Then the Nemytskii operators ${h_{ϵ}^{e}}_{ϵ \in [0, 1]}$ $\begin{array}{} \displaystyle \{h^e_\epsilon\}_{\epsilon\in [0,1]} \end{array}$are well defined from $ℝ \times X^{\frac{s}{2}}$ $\begin{array}{} \displaystyle \mathbb{R}\times X^{\frac s2} \end{array}$into $L^{\frac{2 n}{n + 2 r}} (Ω)$ $\begin{array}{} \displaystyle L^{\frac{2n}{n+2r}}(\Omega) \end{array}$provided that $r \in [\frac{(ρ - 1) (n - 2)}{4}, 1], s \in [r, 1] \cap [\frac{n}{2} - \frac{2}{ρ - 1}, 1]$ $\begin{array}{} xs\displaystyle r\in \left[\frac{(\rho-1)(n-2)}{4},1\right], s \in [r,1]\cap \left[\frac{n}{2}-\frac{2}{\rho-1},1\right] \end{array}$, for each h_ε ∊ 𝒢_ε. If B is a bounded subset of $X^{\frac{s}{2}}$ $\begin{array}{} \displaystyle X^{\frac s2} \end{array}$then there exists a constant C = C(B) > 0 such that

$\sup_{t \in ℝ} ‖ h_{ϵ}^{e} (t, u_{1}) - h_{ϵ}^{e} (t, u_{2}) ‖_{L^{\frac{2 n}{n + 2 r}} (Ω)} ⩽ C ‖ u_{1} - u_{2} ‖_{X^{\frac{s}{2}}}, f o r all ϵ \in [0, 1] .$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\|h^e_\epsilon(t,u_1)-h^e_\epsilon(t,u_2)\|_{L^{\frac{2n}{n+2r}}(\Omega)}\leqslant C\|u_1-u_2\|_{X^{\frac s2}}, \hbox{ for all }\epsilon \in [0,1]. \end{array}$$

Moreover, if r can be taken strictly less than 1 andJ ⊂ ℝ is an arbitrary subset, $h_{ε}^{e}$ $\begin{array}{} \displaystyle h_\varepsilon ^e \end{array}$takes J × B in a precompact set of H⁻¹, for eachε ∊ [0,1].

Proof. Following [10], using hypothesis (H1) we have that

$\begin{matrix} \sup_{t \in ℝ} ‖ h_{ϵ}^{e} (t, u) - h_{ϵ}^{e} (t, v) ‖_{L^{\frac{2 n}{n + 2 r}} (Ω)} ⩽ c {[\int_{Ω} {[| u - v | (1 + | u |^{ρ - 1} + | v |^{ρ - 1})]}^{\frac{2 n}{n + 2 r}}]}^{\frac{n + 2 r}{2 n}} \\ ⩽ \tilde{c} ‖ u - v ‖_{L^{\frac{2 n}{n - 2 r}} (Ω)} (1 + ‖ u ‖_{L^{\frac{n (ρ - 1)}{2 r}} (Ω)}^{ρ - 1} + ‖ v ‖_{L^{\frac{n (ρ - 1)}{2 r}} (Ω)}^{ρ - 1}) \\ ⩽ \bar{c} ‖ u - v ‖_{X^{\frac{s}{2}}} (1 + ‖ u ‖_{X^{\frac{s}{2}}}^{ρ - 1} + ‖ v ‖_{X^{\frac{s}{2}}}^{ρ - 1}) . \end{matrix}$ $$\begin{array}{} \displaystyle \begin{equation} \begin{split} \sup_{t\in\mathbb{R}}\|h_\epsilon^e(t,u)-h_\epsilon^e(t,v)&\|_{L^{\frac{2n}{n+2r}}(\Omega)}\leqslant c\left[ \int_\Omega \left[|u-v|(1+|u|^{\rho-1}+|v|^{\rho-1})\right]^{\frac{2n}{n+2r}}\right]^{\frac{n+2r}{2n}}\\ &\leqslant\tilde c\|u-v\|_{L^{\frac{2n}{n-2r}}(\Omega)}\left(1+\|u\|_{L^{\frac{n(\rho-1)}{2r}}(\Omega)}^{\rho-1}+\|v\|_{L^{\frac{n(\rho-1)}{2r}}(\Omega)}^{\rho-1}\right)\\ &\leqslant\bar c\|u-v\|_{X^{\frac s2}}\left(1+\|u\|_{X^{\frac s2}}^{\rho-1}+\|v\|_{X^{\frac s2}}^{\rho-1}\right). \end{split} \end{equation} \end{array}$$(9)

or any $s \in [r, 1] \cap [\frac{n}{2} - \frac{2}{ρ - 1}, 1]$ $\begin{array}{} \displaystyle s \in [r,1]\cap \left[\frac{n}{2}-\frac{2}{\rho-1},1\right] \end{array}$. The last statement holds since H^−r is compact embedded in H⁻¹ and $L^{\frac{2 n}{n + 2 r}} (Ω) ↪ H^{- r}$ $\begin{array}{} \displaystyle {L^{\frac{{2n}}{{n + 2r}}}}(\Omega )\hookrightarrow{H^{ - r}} \end{array}$H^−r for r < 1.

We define now

$C_{2}^{e} = C_{b}^{0} (ℝ \times H_{0}^{1} (Ω), L^{\frac{2 n}{n + 2}} (Ω))$ $\begin{array}{} \displaystyle \mathscr{C}^e_2=C_b^0(\mathbb{R}\times H^1_0(\Omega),L^{\frac{2n}{n+2}}(\Omega)) \end{array}$ as the set of all continuous functions from $ℝ \times H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle \mathbb{R}\times H^1_0(\Omega) \end{array}$ taking values on $L^{\frac{2 n}{n + 2}} (Ω)$ $\begin{array}{} \displaystyle L^{\frac{2n}{n+2}}(\Omega) \end{array}$ that are bounded on sets ℝ × B, where B is a bounded set of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H^1_0(\Omega) \end{array}$ with metric

$d_{2}^{e} (σ_{1}, σ_{2}) = \sum_{k = 1}^{\infty} 2^{- k} \frac{‖ σ_{1} - σ_{2} ‖_{k}^{e}}{1 + ‖ σ_{1} - σ_{2} ‖_{k}^{e}},$ $$\begin{array}{} \displaystyle d_2^e(\sigma_1,\sigma_2)=\sum_{k=1}^\infty 2^{-k}\frac{\|\sigma_1-\sigma_2\|_k^e}{1+\|\sigma_1-\sigma_2\|_k^e}, \end{array}$$

where if $B^{k} = {u \in H_{0}^{1} (Ω) : ‖ u ‖_{H_{0}^{1} (Ω)} ⩽ k}$ $\begin{array}{} \displaystyle B^k=\{u\in H^1_0(\Omega)\colon \|u\|_{H^1_0(\Omega)}\leqslant k\} \end{array}$ we have

$‖ σ_{1} - σ_{2} ‖_{k}^{e} = \sup_{t \in ℝ} \sup_{u \in B^{k}} ‖ σ_{1} (t, u) - σ_{2} (t, u) ‖_{L^{\frac{2 n}{n + 2}} (Ω)} .$ $$ \|\sigma_1-\sigma_2\|_k^e=\sup\limits_{t\in \mathbb{R}}\sup\limits_{u\in B^k}\|\sigma_1(t,u)-\sigma_2(t,u)\|_{L^{\frac{2n}{n+2}}(\Omega)}. $$

Remark 9.

Here we have that $C_{2}^{e}$ $\begin{array}{} \displaystyle \mathscr{C}_2^e \end{array}$ with the metric $d_{2}^{e}$ $\begin{array}{} \displaystyle d_2^e \end{array}$ is a Frechét space, and a sequence {σ_k}_k∊ℕ converges in $C_{2}^{e}$ $\begin{array}{} \displaystyle C_2^e \end{array}$ if and only if it converges in each seminorm $‖ \cdot ‖_{n}^{e}$ $\begin{array}{} \displaystyle \|\cdot\|_n^e \end{array}$.

Define the group of translations

Again, since there will be no confusion, we denote the group of translations the same.

{θ_t : t ∊ ℝ} in

C_{2}^{e}

$\begin{array}{} \displaystyle \mathscr{C}_2^e \end{array}$

$θ_{t} h (s, v) = h (t + s, v), for all h \in C_{2}^{e}, t, s \in ℝ and v \in H_{0}^{1} (Ω) .$ $$\begin{array}{} \displaystyle \theta_t h(s,v) = h(t+s,v), \hbox{ for all } h\in \mathscr{C}_2^e, \ t,s\in \mathbb{R} \hbox{ and }v\in H^1_0(\Omega). \end{array}$$

Since Lemma 7 implies that $g_{ϵ}^{e} \in C_{2}^{e}$ $\begin{array}{} \displaystyle g_\epsilon^e\in \mathscr{C}_2^e \end{array}$, for all ε ∊ [0,1], let also

$G_{ε}^{e}$ $\begin{array}{} \displaystyle \mathscr{G}^e_\epsilon \end{array}$ be the hull of $g_{ϵ}^{e}$ $\begin{array}{} \displaystyle g^e_\epsilon \end{array}$ in $C_{2}^{e}$ $\begin{array}{} \displaystyle \mathscr{C}_2^e \end{array}$; that is,

$G_{ϵ}^{e} = closure of {θ_{t} g_{ϵ}^{e}}_{t \in ℝ} in (C_{2}^{e}, d_{2}^{e}) .$ $$\begin{array}{} \displaystyle \mathscr{G}^e_\epsilon=\hbox{ closure of } \{\theta_tg^e_\epsilon\}_{t\in \mathbb{R}} \hbox{ in } (\mathscr{C}_2^e,d_2^e). \end{array}$$

Now, in order to have a better understanding of the set $G_{ϵ}^{e}$ $\begin{array}{} \displaystyle \mathscr{G}_\epsilon^e \end{array}$, we present the following results:

Lemma 8.

If h ∊ 𝒢_ε, then $h^{e} \in G_{ϵ}^{e}$ $\begin{array}{} \displaystyle h^e\in \mathscr{G}_\epsilon^e \end{array}$.

Proof. This follows easily by Lemma 7.

Lemma 9.

There exists a constant L ⩾ 0, such that for all h₁,h₂ ∊ 𝒢_ε, we have that

$d_{2}^{e} (h_{1}^{e}, h_{2}^{e}) ⩽ L ‖ h_{1} - h_{2} ‖_{C_{2}} .$ $$\begin{array}{} \displaystyle d_2^e(h^e_1,h^e_2)\leqslant L\|h_1-h_2\|_{\mathscr{C}_2}. \end{array}$$

Proof. Fix k ∊ ℕ. We have that

$\begin{matrix} ‖ h_{1}^{e} - h_{2}^{e} ‖_{k}^{e} = \sup_{t \in ℝ} \sup_{u \in B^{k}} ‖ h_{1}^{e} (t, u) - h_{2}^{e} (t, u) ‖_{L^{\frac{2 n}{n + 2}} (Ω)} \\ = \sup_{t \in ℝ} \sup_{u \in B^{k}} {(\int_{Ω} | h_{1} (t, u (x)) - h_{2} (t, u (x)) |^{\frac{2 n}{n + 2}} d x)}^{\frac{n + 2}{2 n}} \\ ⩽ \tilde{c} ‖ h_{1} - h_{2} ‖_{C_{2}} \cdot \sup_{u \in B^{k}} {(\int_{Ω} | u (x) (1 + | u (x) |^{ρ - 1}) |^{\frac{2 n}{n + 2}} d x)}^{\frac{n + 2}{2 n}}, \end{matrix}$ $$\begin{array}{l} \displaystyle \|h_1^e-&h_2^e\|_k^e=\sup\limits_{t\in \mathbb{R}}\sup\limits_{u\in B^k}\|h_1^e(t,u)-h_2^e(t,u)\|_{L^{\frac{2n}{n+2}}(\Omega)}\\ &=\sup\limits_{t\in \mathbb{R}}\sup\limits_{u\in B^k} \left(\int_\Omega|h_1(t,u(x))-h_2(t,u(x))|^{\frac{2n}{n+2}}dx\right)^{\frac{n+2}{2n}}\\ &\leqslant \tilde{c} \|h_1-h_2\|_{\mathscr{C}_2}\cdot \sup\limits_{u\in B^k}\left(\int_\Omega|u(x)(1+|u(x)|^{\rho-1})|^{\frac{2n}{n+2}}dx\right)^{\frac{n+2}{2n}}, \end{array}$$

where $\tilde{c}$ $\begin{array}{} \displaystyle \tilde{c} \end{array}$ does not depend on h₁,h₂ and hence, arguing as in Lemma 7, we obtain

$\begin{matrix} ‖ h_{1}^{e} - h_{2}^{e} ‖_{k}^{e} ⩽ \bar{c} ‖ h_{1} - h_{2} ‖_{C_{2}} \cdot \sup_{u \in B^{k}} [‖ u ‖_{H_{0}^{1} (Ω)} (1 + ‖ u ‖_{H_{0}^{1} (Ω)}^{ρ - 1})] \\ ⩽ \hat{c} ‖ h_{1} - h_{2} ‖_{C_{2}} [k (1 + k^{ρ - 1})], \end{matrix}$ $$\begin{array}{} \displaystyle \|h_1^e-h_2^e\|_k^e & \leqslant \overline{c} \|h_1-h_2\|_{\mathscr{C}_2}\cdot \sup_{u\in B^k} \big[\|u\|_{H^1_0(\Omega)}(1+\|u\|_{H^1_0(\Omega)}^{\rho -1})\big] \\ & \leqslant \hat{c} \|h_1-h_2\|_{\mathscr{C}_2} [k(1+k^{\rho-1})], \end{array}$$

where ĉ does not depend on h₁,h₂.

Hence

$d_{2}^{e} (h_{1}^{e}, h_{2}^{e}) ⩽ \hat{c} ‖ h_{1} - h_{2} ‖_{C_{2}} \sum_{k = 1}^{\infty} 2^{- k + 1} k^{ρ},$ $$\begin{array}{} \displaystyle d_2^e(h_1^e,h_2^e)\leqslant \hat{c}\|h_1-h_2\|_{\mathscr{C}_2}\sum_{k=1}^{\infty} 2^{-k+1}k^{\rho}, \end{array}$$

and the result follows since $\sum_{k = 1}^{\infty} 2^{- k + 1} k^{ρ}$ $ \sum _{k = 1}^\infty {2^{ - k + 1}}{k^\rho }$ is a convergent series.

Proposition 10.

If condition (C) holds then $σ_{ϵ} \in G_{ϵ}^{e}$ $\begin{array}{} \displaystyle \sigma_\epsilon \in \mathscr{G}_\epsilon^e \end{array}$if and only if there exists h_ε ∊ 𝒢_εsuch that $σ_{ϵ} = h_{ϵ}^{e}$ $\begin{array}{} \displaystyle \sigma_\epsilon=h_\epsilon^e \end{array}$, for eachε ∊ [0,1].

Proof. The result is trivial if ε = 0. Assume that ε ∊ (0,1]. One inclusion follows from Lemma 8. Now if $σ_{ϵ} \in G_{ϵ}^{e}$ $\begin{array}{} \displaystyle \sigma_\epsilon\in \mathscr{G}_\epsilon^e \end{array}$, then exists a sequence {t_n}_n∊ℕ ⊂ ℝ such that $θ_{t_{n}} g_{ϵ}^{e} \to σ_{ϵ}$ $\begin{array}{} \displaystyle \theta_{t_n}g^e_\epsilon \to \sigma_\epsilon \end{array}$ in $C_{2}^{e}$ $\begin{array}{} \displaystyle \mathscr{C}_2^e \end{array}$, by definition. Consider the sequence {θ_{t_n}g_ε}_n∈ℕ in 𝒞₂. Since 𝒢_ε is compact, we can assume without loss of generality, that θ_{t_n}g_ε → h_ε ∊ 𝒢_ε in 𝒞₂. Thus, using Lemma 9 we have that

$\begin{matrix} d_{2}^{e} (σ_{ϵ}, h_{ϵ}^{e}) & ⩽ d_{2}^{e} (σ_{ϵ}, θ_{t_{n}} g_{ϵ}^{e}) + d_{2}^{e} (θ_{t_{n}} g_{ϵ}^{e}, h_{ϵ}^{e}) \\ ⩽ d_{2}^{e} (σ_{ϵ}, θ_{t_{n}} g_{ϵ}^{e}) + L ‖ θ_{t_{n}} g_{ϵ} - h_{ϵ} ‖_{C_{2}}, \end{matrix}$ $$\begin{array}{} \displaystyle d_2^e(\sigma_\epsilon,h_\epsilon^e)&\leqslant d_2^e(\sigma_\epsilon,\theta_{t_n}g^e_\epsilon)+d_2^e(\theta_{t_n}g_\epsilon^e,h_\epsilon^e)\\ &\leqslant d_2^e(\sigma_\epsilon,\theta_{t_n}g^e_\epsilon) + L\|\theta_{t_n}g_\epsilon-h_\epsilon\|_{\mathscr{C}_2}, \end{array}$$

and making n → ∞ we obtain that $σ_{ϵ} = h_{ϵ}^{e}$ $\begin{array}{} \displaystyle \sigma_\epsilon=h_\epsilon^e \end{array}$.

Corollary 11.

If condition (C) holds, then $G_{ϵ}^{e}$ $\begin{array}{} \displaystyle \mathscr{G}_\epsilon^e \end{array}$ is compact in $C_{2}^{e}$ $\begin{array}{} \displaystyle \mathscr{C}_2^e \end{array}$, for eachε ∊ [0,1].

Now we are finally in condition to define the space that will be suitable for our study of (6). Define

$C_{*} = C_{b}^{0} (ℝ \times H_{0}^{1} (Ω), H_{0}^{1} (Ω))$ $\begin{array}{} \displaystyle \mathscr{C}_*=C^0_b(\mathbb{R}\times H^1_0(\Omega),H^1_0(\Omega)) \end{array}$ the set of continuous functions which are bounded in sets of the form ℝ × B, where B is a bounded subset of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H^1_0(\Omega) \end{array}$, with distance defined as

$d_{*} (σ_{1}, σ_{2}) = \sum_{n = 1}^{\infty} 2^{- n} \frac{‖ σ_{1} - σ_{2} ‖_{n}^{*}}{1 + ‖ σ_{1} - σ_{2} ‖_{n}^{*}},$ $$\begin{array}{} \displaystyle d_*(\sigma_1,\sigma_2)=\sum_{n=1}^\infty 2^{-n}\frac{\|\sigma_1-\sigma_2\|_n^*}{1+\|\sigma_1-\sigma_2\|_n^*}, \end{array}$$

where if $B^{n} = {u \in H_{0}^{1} (Ω) : ‖ u ‖_{H_{0}^{1} (Ω)} ⩽ n}$ $\begin{array}{} \displaystyle B^n=\{u\in H^1_0(\Omega)\colon \|u\|_{H^1_0(\Omega)}\leqslant n\} \end{array}$ we have

$‖ σ_{1} - σ_{2} ‖_{n}^{*} = \sup_{t \in ℝ} \sup_{u \in B^{n}} ‖ σ_{1} (t, u) - σ_{2} (t, u) ‖_{H_{0}^{1} (Ω)} .$ $$\begin{array}{} \displaystyle \|\sigma_1-\sigma_2\|_n^*=\sup_{t\in \mathbb{R}}\sup_{u\in B^n}\|\sigma_1(t,u)-\sigma_2(t,u)\|_{H^1_0(\Omega)}. \end{array}$$

Now let $F_{ϵ} (t, u) = - {\tilde{A}}_{γ} (t) u + B_{γ} (t) g^{e} (t, u)$ $\begin{array}{} \displaystyle F_\epsilon(t,u)=-\tilde{A}_\gamma(t)u+B_\gamma(t)g^e(t,u) \end{array}$ be given as in (8). It is simple to see, recalling the definitions of ${\tilde{A}}_{γ} (t)$ $\begin{array}{} \displaystyle \tilde{A}_\gamma(t) \end{array}$ and B_γ(t) and the fact that $g_{ϵ}^{e} \in C_{2}^{e}$ $\begin{array}{} \displaystyle g_\epsilon^e\in \mathscr{C}_2^e \end{array}$, that F_ε ∊ 𝒞_* for each ε ∊ [0,1]. Again, we can define the group

^a We once again denote the same

{θ_t: t ∊ ℝ} in 𝒞_* by

$θ_{t} h (s, v) = h (t + s, v), for all h \in C_{*}, t, s \in ℝ and v \in H_{0}^{1} (Ω) .$ $$\begin{array}{} \displaystyle \theta_t h(s,v)=h(t+s,v), \hbox{ for all } h\in \mathscr{C}_*, \ t,s\in \mathbb{R} \hbox{ and } v\in H^1_0(\Omega). \end{array}$$

Definition 10

With the notations above, we set Σ_ε as the hull of F_ε in 𝒞_*; that is,

$Σ_{ϵ} = closure of {θ_{t} F_{ϵ}}_{t \in ℝ} in (C_{*}, d_{*}) .$ $$\begin{array}{} \displaystyle \Sigma_\epsilon= \hbox{ closure of } \{\theta_tF_\epsilon\}_{t\in \mathbb{R}} \hbox{ in } (\mathscr{C}_*,d_*). \end{array}$$

Before we proceed with the study of (6), we will need some characterization result for Σ_ε.

Lemma 12

We have that

(a) $Σ_{0} = {B_{λ} f^{e} - {\tilde{A}}_{λ}}_{λ \in Γ}$ $\begin{array}{} \displaystyle {\Sigma _0} = {\{ {B_\lambda }{f^e} - {\tilde A_\lambda }\} _{\lambda \in \Gamma }} \end{array}$and it is compact in (𝒞_*,d_*);

if(C)holds true, then for each ε > 0 we have

$Σ_{ϵ} \subseteq {B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}}_{λ \in Γ, h ϵ \in G_{ϵ}},$ $$\begin{array}{} \displaystyle \Sigma_\epsilon\subseteq \{B_\lambda h_\epsilon^e - \tilde{A}_\lambda\}_{\lambda\in \Gamma,h\epsilon\in \mathscr{G}_\epsilon}, \end{array}$$

and Σ_ε is compact in (𝒞_*,d_*).

where B_λ(t) = (I + λ(t)A)^–1and ${\tilde{A}}_{λ} (t) = A B_{λ} (t)$ $\begin{array}{} \displaystyle {\tilde A_\lambda }(t) = A{B_\lambda }(t) \end{array}$, for each λ ∈ Γ.

Proof. Since g₀(t,s) = f(s) for all t,s ∈ ℝ and Γ is compact, item (a) follows immediately. Now, fix ε > 0 and let H_ε ∈ Σ_ε. Then, by definition, there exists a real sequence {t_n}_n∈ℕ such that $H_{ϵ} = C_{*} - \lim_{n \to \infty} θ_{t_{n}} (B_{γ} g_{ϵ}^{e} - {\tilde{A}}_{γ})$ $\begin{array}{} \displaystyle H_\epsilon=\mathscr{C}_*-\displaystyle \lim_{n\to \infty}\theta_{t_n}(B_\gamma g^e_\epsilon - \tilde{A}_\gamma) \end{array}$ that is, the sequence $θ_{t_{n}} (B_{γ} g_{ϵ}^{e} - {\tilde{A}}_{γ})$ $\begin{array}{} \displaystyle \theta_{t_n}(B_\gamma g^e_\epsilon - \tilde{A}_\gamma) \end{array}$ converges to H_ε in the metric of 𝒞_* defined above.

We can extract a subsequence of {t_n}_n∈ℕ, which we shall denote the same, and elements λ ∈ Γ and h_ε ∈ 𝒢_ε such that $γ = C - \lim_{n \to \infty} θ_{s_{n}} γ$ $\begin{array}{} \displaystyle \gamma = {\cal C} - \mathop {\lim }\limits_{n \to \infty } {\theta _{{s_n}}}\gamma \end{array}$ and $h_{ϵ}^{e} = C_{2}^{e} - \lim_{n \to \infty} θ_{t_{n}} g_{ϵ}^{e}$ $\begin{array}{} \displaystyle h_\epsilon^e=\mathscr{C}_2^e-\displaystyle\lim_{n\to \infty}\theta_{t_n}g_\epsilon^e \end{array}$, by the compactness of Γ and 𝒢_ε and Proposition 10. Thus $H_{ϵ} = B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_\epsilon=B_\lambda h_\epsilon^e -\tilde{A}_\lambda \end{array}$.

The last statement is clear from (a) and (b), which concludes the result.

Thus our problem in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$ takes the form

${\begin{matrix} \dot{u} = H_{ϵ} (t, u), t > 0 \\ u (0) = u_{0} \in H_{0}^{1} (Ω), \end{matrix}$ $$\left\{\begin{array}{l} &\dot{u}=H_\epsilon(t,u), \quad t \gt 0\\ &u(0)=u_0\in H^1_0(\Omega), \end{array}\right.$$(10)

for each H_ε ∈ Σ_ε, which is precisely equation (4). As a matter of fact we have, for each fixed ε ∈ [0,1], a problem equals to (2). Our first task is to find, for each ε ∈ [0,1] and H_ε ∈ Σ_ε, a solution t ↦ φ(t, H_ε)u₀ of (10). But we will solve this problem in a slightly different way, considering all possible functions in ${B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}}_{λ \in Γ, h_{ϵ} \in G_{ϵ}}$ $\begin{array}{} \displaystyle \{B_\lambda h^e_\epsilon-\tilde{A}_\lambda\}_{\lambda\in \Gamma,h_\epsilon\in \mathscr{G}_\epsilon} \end{array}$ and therefore, we will denote this space by Γ ⋄ 𝒢_ε; that is,

$Γ ⋄ G_{ϵ} = {B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}}_{λ \in Γ, h_{ϵ} \in G_{ϵ}} .$ $$\begin{array}{} \displaystyle \Gamma\diamond \mathscr{G}_\epsilon=\{B_\lambda h^e_\epsilon-\tilde{A}_\lambda\}_{\lambda\in \Gamma,h_\epsilon\in \mathscr{G}_\epsilon}. \end{array}$$(11)

Remark 10.

It is clear that the maps ℝ ∋ t ↦ B_λ(t) and $ℝ ∋ t \mapsto {\tilde{A}}_{λ} (t)$ $\begin{array}{} \displaystyle \mathbb{R}\ni t\mapsto \tilde{A}_\lambda(t) \end{array}$ are absolutely continuous, for each λ ∈ Γ.

Non-autonomous dynamical systems and skew-product semiflows for (10)

In this section we will show that equation (10) generates a non-autonomous dynamical system ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$, for each ε ∈ [0,1].

4.1

Local existence and uniqueness of solutions

Using Remark 10, Lemma 7 and the results on [24, Chapter 5] we have the following theorem of local existence and uniqueness of solutions.

Theorem 13.

Assume that hypotheses (H1) and (C) are satisfied. Then, for each bounded subset $B \subset H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle B \subset H_0^1(\Omega ) \end{array}$, ε ∈ [0,1] andH_ε ∈ Γ ⋄ 𝒢_εthere exists ε = ε(B, ε, H_ε) > 0 such that for each u₀ ∈ B there exists a unique function

$[0, ε] ∋ t \mapsto φ_{ϵ} (t, H_{ϵ}) u_{0},$ $$\begin{array}{} \displaystyle [0,\omega]\ni t\mapsto \varphi_\epsilon(t,H_\epsilon)u_0, \end{array}$$

with $φ_{ϵ} (\cdot, H_{ϵ}) u_{0} \in C^{1} ([0, ε], H_{0}^{1} (Ω))$ $\begin{array}{} \displaystyle \varphi_\epsilon(\cdot,H_\epsilon)u_0\in C^1([0,\omega],H^1_0(\Omega)) \end{array}$satisfying (10); that is, φ_ε(0,H_ε)u₀ = u₀ and

$\begin{matrix} \frac{d}{d t} φ_{ϵ} (t, H_{ϵ}) u_{0} = H_{ϵ} (t, φ_{ϵ} (t, H_{ϵ}) u_{0}), for 0 < t < ω . \end{matrix}$ $$\begin{array}{l} \frac{d}{dt}\varphi_\epsilon(t,H_\epsilon)u_0 = H_\epsilon(t,\varphi_\epsilon(t,H_\epsilon)u_0), \;\text{for} \;0 \lt t \lt \omega. \end{array}$$

4.2

Global existence of solutions

Assume that (H2) holds true. Following the ideas of [19,26], for any $v \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle v\in H^1_0(\Omega) \end{array}$, h_ε ∈ 𝒢_ε and each δ > 0 there exists a constant K_δ > 0 such that

$\begin{matrix} \int_{Ω} h_{ϵ}^{e} (t, v) v ⩽ δ ‖ v ‖_{L^{2} (Ω)}^{2} + K_{δ}, \\ \int_{Ω} Φ_{ϵ} (t, v) ⩽ δ ‖ v ‖_{L^{2} (Ω)}^{2} + K_{δ} \end{matrix}$ $$\begin{array}{} \displaystyle &\int_\Omega h^e_\epsilon(t,v) v \leqslant \delta\|v\|_{L^2(\Omega)}^2 + K_\delta,\\ &\int_\Omega \Phi_\epsilon(t,v) \leqslant \delta\|v\|_{L^2(\Omega)}^2 + K_\delta \end{array}$$(12)

with $Φ_{ϵ} (t, r) = \int_{0}^{r} h_{ϵ} (t, θ) d θ$ $\begin{array}{} \displaystyle \Phi_\epsilon(t,r)=\int_0^r h_\epsilon(t,\theta)d\theta \end{array}$, uniformly in t ∈ ℝ and ε ∈ [0,1].

Now for each $v \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle v \in H_0^1(\Omega ) \end{array}$, we define the energy functional L_b,ε(t, v) as

$L_{b, ϵ} (t, v) = \frac{1}{2} (‖ v ‖_{L^{2} (Ω)}^{2} + b ‖ v ‖_{H_{0}^{1} (Ω)}^{2}) - b \int_{Ω} Φ_{ϵ} (t, v),$ $$\begin{array}{} \displaystyle L_{b,\epsilon}(t,v)=\frac12\left(\|v\|^2_{L^2(\Omega)}+b\|v\|^2_{H^1_0(\Omega)}\right)-b\int_\Omega \Phi_\epsilon(t,v), \end{array}$$(13)

with b > 0. It is easy to prove that for $δ ⩽ \frac{1}{2 b}$ $\begin{array}{} \displaystyle \delta \leqslant \frac{1}{{2b}} \end{array}$,

$L_{b, ϵ} (t, v) ⩾ \frac{b}{2} ‖ v ‖_{H_{0}^{1} (Ω)}^{2} - b K_{δ}$ $$\begin{array}{} \displaystyle {L_{b,}}(t,v)\geqslant \frac{b}{2}v_{H_0^1(\Omega )}^2 - b{K_\delta } \end{array}$$(14)

and for any δ > 0,

$L_{b, ϵ} (t, v) ⩽ \frac{b λ_{1} + 2 (1 + b δ)}{2 λ_{1}} ‖ v ‖_{H_{0}^{1} (Ω)}^{2} + b K_{δ},$ $$\begin{array}{} \displaystyle {L_{b,}}(t,v)\leqslant \frac{{b{\lambda _1} + 2(1 + b\delta )}}{{2{\lambda _1}}}\| v \|_{H_0^1(\Omega )}^2 + b{K_\delta }, \end{array}$$(15)

uniformly in time t ∈ ℝ, with λ₁ > 0 the first eigenvalue of A = −Δ with Dirichlet boundary conditions.

Now, assuming that (H3) holds true and using (12), for a solution u(t) = φ_ε(t, H_ε)u₀ of (10), where H_ε ∈ Γ ⋄ 𝒢_ε, we have

$\begin{matrix} \frac{d}{d t} L_{b, ϵ} (t, u) & ⩽ - λ (t) (u, u_{t})_{H_{0}^{1} (Ω)} - ∥ u ∥_{H_{0}^{1} (Ω)}^{2} \\ + (u, h_{ϵ}^{e} (t, u))_{L^{2} (Ω)} - b (∥ u_{t} ∥_{L^{2} (Ω)}^{2} + λ (t) ∥ u_{t} ∥_{H_{0}^{1} (Ω)}^{2}) + C \\ ⩽ - (1 - \frac{λ_{1} γ_{1} η + 2 δ}{2 λ_{1}}) ∥ u ∥_{H_{0}^{1} (Ω)}^{2} + λ (t) \frac{1 - 2 η b}{2 η} ∥ u_{t} ∥_{H_{0}^{1} (Ω)}^{2} + C, \end{matrix}$ $$\begin{array}{l} \frac{d}{dt}L_{b,\epsilon}(t,u) &\leqslant-\lambda(t)(u,u_t)_{H_0^1(\Omega)}-\|u\|^2_{H_0^1(\Omega)}\\&\qquad +(u,h^e_\epsilon(t,u))_{L^2(\Omega)}-b\left(\|u_t\|^2_{L^2(\Omega)}+\lambda(t)\|u_t\|^2_{H_0^1(\Omega)} \right)+C\\ &\leqslant -\left(1-\frac{\lambda_1\gamma_1\eta+2\delta}{2\lambda_1} \right)\|u\|^2_{H_0^1(\Omega)}+\lambda(t)\frac{1-2\eta b}{2\eta}\|u_t\|^2_{H_0^1(\Omega)}+C, \end{array}$$

for δ, η > 0 and taking $δ \in (λ_{1} - \frac{γ_{1}}{2}, λ_{1}), η < \frac{2 (λ_{1} - δ)}{λ_{1} γ_{1}} and b > \frac{1}{2 η}$ $\delta \in \left( {{\lambda _1} - \frac{{{\gamma _1}}}{2},{\lambda _1}} \right),\eta \lt \frac{{2({\lambda _1} - \delta )}}{{{\lambda _1}{\gamma _1}}}\;{\text{and}\; b} \gt \frac{1}{{2\eta }}$, we have

$\frac{d}{d t} L_{b, ϵ} (t, u) ⩽ - k L_{b, ϵ} (t, u) + C,$ $$\begin{array}{} \displaystyle \frac{d}{{dt}}{L_{b,}}(t,u) \leqslant - k{L_{b,}}(t,u) + C, \end{array}$$

with k, C > 0 that do not depend neither on time t ∈ ℝ nor on ε ∈ [0,1]. Therefore,

$‖ u (t) ‖_{H_{0}^{1} (Ω)}^{2} ⩽ K ‖ u_{0} ‖_{H_{0}^{1} (Ω)}^{2} e^{- k t} + C,$ $$\begin{equation} \|u(t)\|^2_{H_0^1(\Omega)}\leqslant K\|u_0\|^2_{H_0^1(\Omega)}e^{-kt}+C, \end{equation}$$(16)

for certain constants k > 0 and K, C ⩾ 0 which do not depend on time, ε and H_ϵ ∈ Γ ⋄ 𝒢_ϵ, and thus we have the global existence of solutions for (10). To summarise, we have so far the following result

Theorem 14

Assume that conditions(H1)-(H3)and(C)are satisfied. Then, for each ε ∈ [0,1], H_ϵ ∈ Γ ⋄ 𝒢_ϵand $u_{0} \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {u_0} \in H_0^1(\Omega ) \end{array}$, equation (10) has a solution φ (·, H_ε)u₀ defined for all t ⩾ 0. Moreover, there exists abounded subset B₀ of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$, independent of ε ∈ [0,1], such that for each bounded subset $B \subset H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle B \subset H_0^1(\Omega ) \end{array}$, there exists T = T (B) ⩾ 0 such that for t ⩾ T we have

$φ_{ϵ} (t, H_{ϵ}) B \subset B_{0} .$ $$\begin{array}{} \displaystyle \varphi_\epsilon(t,H_\epsilon)B\subset B_0. \end{array}$$

Proof. It is simple to see that, using (16), each solution given by Theorem 13 exists for all t ⩾ 0. Now define

$B_{0} = {v \in H_{0}^{1} (Ω) : ‖ v ‖_{H_{0}^{1} (Ω)}^{2} ⩽ 2 C},$ $$ B_0=\{v\in H^1_0(\Omega)\colon \|v\|^2_{H^1_0(\Omega)}\leqslant 2C\}, $$

where C is given in (16). Given B a bounded subset of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$, set $∥ B ∥ = sup_{v \in B} ∥ v ∥_{H_{0}^{1} (Ω)}$ $\|B\|=\sup_{v\in B}\|v\|_{H^1_0(\Omega)}$ and choose

$T = \frac{1}{k} \ln (\frac{K ‖ B ‖^{2}}{C}),$ $$ T=\frac1k\ln\left(\frac{K\|B\|^2}{C}\right), $$

where k and K are given in (16).

Since Σ_ε ⊂ Γ ⋄ 𝒢_ε and Σ_ε is a compact invariant set for the group of translations {θ_t: t ∈ ℝ} in 𝒞_*, a simple consequence of Theorem 14 is the following:

Theorem 15

Assume that conditions(H1)-(H3)and(C)are satisfied. Then, for each ε ∈ [0,1], equation (10) generates a non-autonomous dynamical system (φ_ε, θ) in $(H_{0}^{1} (Ω), Σ_{ϵ})$ $\begin{array}{} \displaystyle (H^1_0(\Omega),\Sigma_\epsilon) \end{array}$, where for each ε ∈ [0,1], H_ε ∈ Σ and $u_{0} \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {u_0} \in H_0^1(\Omega ) \end{array}$, the function ℝ₊ ∋ t ↦ φ_ε (t, H_ε)u₀is the unique solution of (10).

4.3

Skew-product semiflows for (10)

Using (3), we are able to define, for each ε ∈ [0,1], the skew-product semiflow

${Π_{ϵ} (t) : t ⩾ 0} in X_{ϵ} = H_{0}^{1} (Ω) \times Σ_{ϵ}$ $$\begin{array}{} \displaystyle \{\Pi_\epsilon(t)\colon t \geqslant 0\} \hbox{ in } \mathbb{X}_\epsilon= H^1_0(\Omega)\times \Sigma_\epsilon \end{array}$$(17)

associated with ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$, by setting

$Π_{ϵ} (t) (u_{0}, H_{ϵ}) = (φ_{ϵ} (t, H_{ϵ}) u_{0}, θ_{t} H_{ϵ}),$ $$\begin{array}{} \displaystyle \Pi_\epsilon(t)(u_0,H_\epsilon)=(\varphi_\epsilon(t,H_\epsilon)u_0,\theta_tH_\epsilon), \end{array}$$

for all t ⩾ 0, (u₀, H_ε) ∊ ℝ_ε and ε ∈ [0,1].

Different attractors for (10)

In this section we will use the result of Section 2 to obtain different attractors for equation (10), for the different frameworks described in Subsection 1.1.

5.1

Global attractors for skew-product semiflows

In this section, we will use the theory of autonomous equations to find a global attractor for each one of the skew-product semiflows defined by (17).

We first will see a very simple result, that follows from Theorem 14.

Proposition 16.

Assume that(H1)-(H3)and(C)are satisfied and fix ε ∈ [0,1]. Then there exists a bounded set $B_{ϵ} \subset X_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{B}_\epsilon\subset \mathbb{X}_\epsilon \end{array}$such that given a bounded subset $B \subset X_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{B}\subset \mathbb{X}_\epsilon \end{array}$there exists T = T (B) ⩾ 0 such that

$\begin{matrix} Π_{ϵ} (t) B \subset B_{ϵ}, f o r a l l t ⩾ T . \end{matrix}$ $$\begin{array}{l} \displaystyle \Pi_\epsilon(t)\mathbb{B}\subset \mathbb{B}_\epsilon, {for \; all\; }t \geqslant T. \end{array}$$

Proof. Let $B_{0} \subset H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {B_0} \subset H_0^1(\Omega ) \end{array}$ be as in Theorem (14) and define $B_{ϵ} ≐ B_{0} \times Σ_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{B}_\epsilon\doteq B_0\times \Sigma_\epsilon \end{array}$. Since B₀ is bounded in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$ and Σ is compact space, we have that 𝔹_ε is bounded in 𝕏_ε. Moreover, if 𝔹 is a bounded subset of 𝕏_ε, we have that $B \subset B \times Σ_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{B}\subset B\times \Sigma_\epsilon \end{array}$, for some bounded set $B \subset H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle B \subset H_0^1(\Omega ) \end{array}$. Hence, if T is as in Theorem 14 we obtain the result.

5.2

Asymptotical compactness

In order to obtain a global attractor for each skew-product semiflow, we must prove that the semigroups {Π_ε(t): t ⩾ 0} are asymptotically compact. That is our goal for the next few results.

Lemma 17

Define $H_{λ} (t, v) = - {\tilde{A}}_{λ} (t) v \in C_{*}$ $\begin{array}{} \displaystyle {H_\lambda }(t,v) = - {\tilde A_\lambda }(t)v \in {{\cal C}_*} \end{array}$. Then the problem

${\begin{matrix} \dot{u} = H_{λ} (t, u), t > 0 \\ u (0) = u_{0} \in H_{0}^{1} (Ω) \end{matrix}$ $$ \left\{\begin{array}{*{20}{l}} {\dot u = {H_\lambda }(t,u),\;t \gt 0}\\ {u(0) = {u_0} \in H_0^1(\Omega )} \end{array}\right.$$

has a unique solution φ_χ(t) defined for allt ⩾ 0 given by

$\begin{matrix} φ_{λ} (t) u_{0} = u_{0} - \int_{0}^{t} {\tilde{A}}_{λ} (s) φ_{λ} (s) u_{0} d s, f o r a l l t ⩾ 0. \end{matrix}$ $$\begin{array}{l} \displaystyle {\varphi _\lambda }(t){u_0} = {u_0} - \int_0^t {{{\tilde A}_\lambda }} (s){\varphi _\lambda }(s){u_0}\;ds,{\;{ for \;all }\;}t \geqslant 0. \end{array}$$(18)

and also, there exists constants K, k > 0 which do not depend on λ such that

$∥ φ_{λ} (t) u_{0} ∥_{H_{0}^{1} (Ω)} ⩽ K ∥ u_{0} ∥_{H_{0}^{1} (Ω)} e^{- k t}, f o r a l l t ⩾ 0.$ $$\begin{equation} \|\varphi_\lambda(t)u_0\|_{H^1_0(\Omega)}\leqslant K\|u_0\|_{H^1_0(\Omega)}e^{-kt}, \;{ for \;all }\; t\geqslant 0. \end{equation}$$(19)

Proof. The local existence and uniqueness of φ_* follows from Theorem 13. Proceeding as in Subsection 4.2 with h_ϵ ≡ 0, we can see that we can take C = 0 in (16) and gives us the global existence of φ_* and (19). Equation (18) is a simple consequence of the theory of ordinary differential equations, since $- {\tilde{A}}_{λ} (t)$ $\begin{array}{} \displaystyle - {\tilde A_\lambda }(t) \end{array}$ is a uniformly bounded operator of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$, and the bounds do not depend on λ ∈ Γ.

Lemma 18

If $H_{ϵ} = B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ} \in Σ_{ϵ}$ $\begin{array}{} \displaystyle H_\epsilon=B_\lambda h^e_\epsilon-\tilde{A}_\lambda \in \Sigma_\epsilon \end{array}$ and $u_{0} \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle u_0\in H^1_0(\Omega) \end{array}$then

$φ_{ϵ} (t, H_{ϵ}) u_{0} = φ_{λ} (t) u_{0} + ψ (t) (u_{0}, H_{ϵ}), f o r e a c h t ⩾ 0,$ $$\begin{array}{} \displaystyle \varphi_\epsilon(t,H_\epsilon)u_0=\varphi_\lambda(t)u_0+\psi(t)(u_0,H_\epsilon), \;{ for \;each }\; t \geqslant 0, \end{array}$$

where $ψ (t) (u_{0}, H_{ϵ}) ≐ \int_{0}^{t} B_{λ} (s) h_{ϵ}^{e} (s, φ_{ϵ} (s, H_{ϵ}) u_{0}) d s$ $\begin{array}{} \displaystyle \psi(t)(u_0,H_\epsilon)\doteq \int_0^tB_{\lambda}(s)h_\epsilon^e(s,\varphi_\epsilon(s,H_\epsilon)u_0)ds \end{array}$. Moreover, themap ψ(t) is a compact map from $H_{0}^{1} (Ω) \times Σ_{ϵ}$ $\begin{array}{} \displaystyle H^1_0(\Omega)\times \Sigma_\epsilon \end{array}$into $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$.

Proof. Clearly the right side of the equation is a solution of (10), thus the uniqueness shows the equality. Now let B be a bounded set of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$. By (16) the set $B ≐ {φ_{ϵ} (s, H_{ϵ}) B : s \in [0, t], H_{ϵ} \in Σ_{ϵ}}$ $\begin{array}{} \displaystyle \mathscr{B}\doteq \{\varphi_\epsilon(s,H_\epsilon)B\colon s\in[0,t], \ H_\epsilon\in \Sigma_\epsilon\} \end{array}$ is bounded and thus Lemma 7 ensures that $\cup_{h_{ϵ} \in G_{ϵ}} h_{ϵ}^{e} ([0, t], B)$ $\begin{array}{} \displaystyle \cup_{h_\epsilon\in \mathscr{G}_\epsilon}h_\epsilon^e([0,t],\mathscr{B} \end{array}$ is a precompact set of H^—1. The fact that B_λ (t) is a uniformly bounded bounded linear operator for t ∈ ℝ and λ ∈ Γ concludes the proof.

Using these two lemmas, we are able to prove the asymptotical compactness for the skew-product semiflows.

Proposition 19

The skew-product semiflow {Π_ϵ(t): t ⩾ 0} is asymptotically compact, for each ε ∈ [0,1].

Proof. Let {u_n}_n∈ℕ be a bounded sequence in $H_{0}^{1} (Ω), {H_{ϵ, n}}_{n \in ℕ}$ $\begin{array}{} \displaystyle H_0^1(\Omega ),{\{ {H_{,n}}\} _{n \in \mathbb{N}}} \end{array}$ a bounded sequence in Σ_ε and {t_n}_n∈ℕ be a real sequence with t_n → ∞ as n → ∞ and such that the sequence {Π_ε(t_n)(u_n, H_ε,n)} is bounded. We have

$Π_{ϵ} (t_{n}) (u_{n}, H_{ϵ, n}) = (φ_{ϵ} (t_{n}, H_{ϵ, n}) u_{n}, θ_{t_{n}} H_{ϵ, n}), for all n \in N .$ $$ \Pi_\epsilon(t_n)(u_n,H_{\epsilon,n})=(\varphi_\epsilon(t_n,H_{\epsilon,n})u_n,\theta_{t_n}H_{\epsilon,n}), \hbox{ for all } n\in \mathbb{N}. $$

Since Σ is compact, we can assume, up to a subsequence, that there exists H_ε ε Σ_ε such that $H_{ϵ} = C_{*} - \lim_{n \to \infty} θ_{t_{n}} H_{ϵ, n}$ $\begin{array}{} \displaystyle H_{\epsilon}=\mathscr{C}_*-\displaystyle \lim_{n\to \infty}\theta_{t_n}H_{\epsilon,n} \end{array}$.

Now using Lemma 18 we can write

$φ_{ϵ} (t_{n}, H_{ϵ, n}) u_{n} = φ_{*} (t_{n}) u_{n} + ψ (t_{n}) (u_{n}, H_{ϵ, n}), for each n \in ℕ .$ $$\begin{array}{} \displaystyle \varphi_\epsilon(t_n,H_{\epsilon,n})u_n=\varphi_*(t_n)u_n+\psi(t_n)(u_n,H_{\epsilon,n}), \hbox{ for each } n\in \mathbb{N}. \end{array}$$

Since {u_n}_n∈ℕ is bounded in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$, Lemma 17 implies that φ_*(t_n)u_n → 0 as n → ∞. It is now simple to see that the sequence {φ_ε(t_n,H_ε,n)u_n}_n∈ℕ is precompact in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$, which proves that the sequence {Π_ε(t_n)(u_n, H_ε,n)} has a convergent subsequence in 𝕏_ε and concludes the proof.

Now we can join the results of Propositions 16 and 19, together with Theorem 1, to obtain the next theorem.

Theorem 20

(Existence of the global attractor). Assume that(H1)-(H3)and(C)hold. Then the skew-product semiflow {Π_ε(t): t ⩾ 0} associated with the non-autonomous dynamical system ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ)}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma)} \end{array}$has a global attractor 𝔸_εin 𝕏_εfor each ε ∈ [0,1]. Moreover

$\begin{matrix} A_{ϵ} \subset B_{0} \times Σ_{ϵ}, f o r e a c h ε \in [0, 1], \end{matrix}$ $$\begin{array}{} \displaystyle \mathbb{A}_\epsilon\subset B_0\times \Sigma_\epsilon,\;{for\;each}\; \varepsilon\in [0,1], \end{array}$$

where B₀ is the bounded set given in Theorem 14.

We know that the attractors 𝔸_ε can be characterized by

$\begin{matrix} A_{ϵ} = {(u_{0}, H_{ϵ}) \in X_{ϵ} : & there exists a bounded global solution η_{ϵ} \\ of {Π_{ϵ} (t) : t ⩾ 0} through (u_{0}, H_{ϵ})} . \end{matrix}$ $$\begin{array}{} \displaystyle \mathbb{A}_\epsilon=\{(u_0,H_\epsilon)\in \mathbb{X}_\epsilon\colon &\hbox{ there exists a bounded global solution } \eta_\epsilon \\& \hbox{ of } \{\Pi_\epsilon(t)\colon t \geqslant 0\} \hbox{ through } (u_0,H_\epsilon)\}. \end{array}$$

It is not difficult to see that we can write η_ε as

$η_{ϵ} (t) = (ξ_{ϵ} (t), θ_{t} H_{ϵ}),$ $$\begin{array}{} \displaystyle \eta_\epsilon(t)=(\xi_\epsilon(t),\theta_tH_\epsilon), \end{array}$$

where ξ is a global solution of (10); that is, φ_ε(t – s, θ_sH_ε)ξ_ε(s) = ξ_ε(t) for all t ⩾ s, and ξ (0) = u₀. Using Lemma 18 we can write

$ξ_{ϵ} (t) = φ_{ϵ} (t - s, θ_{s} H_{ϵ}) ξ_{ϵ} (s) = φ_{λ} (t - s) ξ_{ϵ} (s) + ψ (t - s) (ξ_{ϵ} (s), θ_{s} H_{ϵ}),$ $$\begin{array}{} \displaystyle \xi_\epsilon(t)=\varphi_\epsilon(t-s,\theta_sH_\epsilon)\xi_\epsilon(s)=\varphi_\lambda(t-s)\xi_\epsilon(s)+\psi(t-s)(\xi_\epsilon(s),\theta_sH_\epsilon), \end{array}$$(20)

where

$\begin{matrix} ψ (t - s) (ξ_{ϵ} (s), θ_{s} H_{ϵ}) = & \int_{0}^{t - s} B_{θ_{s} λ} (r) θ_{s} h_{ϵ}^{e} (r, φ (r, θ_{s} H_{ϵ}) ξ_{ϵ} (s)) d r \\ = & \int_{0}^{t - s} B_{λ} (r + s) h_{ϵ}^{e} (r + s, ξ_{ϵ} (r + s)) d r \\ = & \int_{s}^{t} B_{λ} (r) h_{ϵ}^{e} (r, ξ_{ϵ} (r)) d r . \end{matrix}$ $$\begin{array}{} \displaystyle \psi(t-s)(\xi_\epsilon(s),\theta_sH_\epsilon)&=\int_0^{t-s}B_{\theta_s\lambda}(r)\theta_sh_\epsilon^e(r,\varphi(r,\theta_sH_\epsilon)\xi_\epsilon(s))dr \\&=\int_0^{t-s}B_\lambda(r+s)h^e_\epsilon(r+s,\xi_\epsilon(r+s))dr\\ &=\int_s^tB_\lambda(r)h_\epsilon^e(r,\xi_\epsilon(r))dr. \end{array}$$

Now, making s → — ∞ in (20), since {ξ_ε(s)}_s∈ℝ is bounded in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$ we obtain, using (19), that

$ξ (t) = \int_{- \infty}^{t} B_{λ} (r) h_{ϵ}^{e} (r, ξ (r)) d r .$ $$\begin{array}{} \displaystyle \xi(t)=\int_{-\infty}^tB_\lambda(r)h_\epsilon^e(r,\xi(r))dr. \end{array}$$

Using Lemma 7 we can show (following the ideas of [26]) that the solution ξ_ε is bounded in $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {H^2}(\Omega ) \cap H_0^1(\Omega ) \end{array}$, and the bound does not depend on the particular ξ_ε (neither it does on ε). Hence we obtain that there exists a bounded subset D₀ of $H^{2} (Ω) \cap H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {H^2}(\Omega ) \cap H_0^1(\Omega ) \end{array}$ such that

$A_{ϵ} \subset D_{0} \times Σ_{ϵ}, for each ϵ \in [0, 1] .$ $$\begin{array}{} \displaystyle \mathbb{A}_\epsilon \subset D_0\times \Sigma_\epsilon, \hbox{ for each } \epsilon\in[0,1]. \end{array}$$(21)

5.3

Other attractors

Now using Theorem 20 we are able to obtain other attractors for equation (10).

Theorem 21.

Assume that(H1)-(H3)and(C)hold true. Then we have, for each ε ∈ [0,1], that

(a)the non-autonomous dynamical system ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$ has a uniform attractor 𝒜_∈ and

$A_{ϵ} = π_{H_{0}^{1} (Ω)} A_{ϵ} \subset D_{0};$ $$\begin{array}{} \displaystyle \mathscr{A}_\epsilon =\pi_{H^1_0(\Omega)}\mathbb{A}_\epsilon \subset D_0; \end{array}$$

(b)the the non-autonomous dynamical system ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$ has a cocycle attractor {A(H_∈)}H_∈∈Σ_ε with ∪_{H_ε}∈ Σ_εA(H_ε) ⊂ D₀, and

$A (H_{ϵ}) = {v \in H_{0}^{1} (Ω) : (v, H_{ϵ}) \in A_{ϵ}};$ $$\begin{array}{} \displaystyle A(H_\epsilon)=\{v\in H^1_0(\Omega)\colon (v,H_\epsilon)\in \mathbb{A}_\epsilon\}; \end{array}$$

(c)for each H_ε ∈ Σ_ε, the evolution process {T_He (t, s): t ⩾ s} given by

T_{H_ε} (t, s) = φ_ε (t — s, θ_sH_ε), for all t ⩾ s,

has a pullback attractor {A_{H_ε}(t)}_{t ∈ ℝ} with ∪_{t ∈ ℝ}A_{H_e}(t) ⊂ D₀, and

$A_{ϵ} = \underset{H_{ϵ} \in Σ_{ϵ}}{\cup} [\underset{t \in ℝ}{\cup} A_{H_{ϵ}} (t) \times {H_{ϵ}}] .$ $$\begin{array}{} \displaystyle \mathbb{A}_\epsilon=\bigcup_{H_\epsilon\in \Sigma_\epsilon}\left[\bigcup_{t\in \mathbb{R}}A_{H_\epsilon}(t)\times \{H_\epsilon\}\right]. \end{array}$$

where D₀is given in (21).

Proof. Using Theorem 20, we have easily that item (a) follows from Theorem 5, item (b) from Theorem 3 and item (c) from Theorem 4.

Upper semicontinuity of attractors

This section is devoted to study the upper semicontinuity of the semigroups {Π_ε(t): t ⩾ 0} as perturbations of {Π₀(t): t ⩾ 0}, as a part of the study described in Subsection 1.2. So far, we have treated the family of equations (6) (and equivalently, (10)) individually for each ε ∈ [0,1], but now it is time to look at all these equations together at once.

Assuming that (H1)-(H3) and (C) hold, we obtained so far a family of semigroups {Π_ε(t): t ⩾ 0} in $X_{ϵ} = H_{0}^{1} (Ω) \times Σ_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{X}_\epsilon=H^1_0(\Omega)\times \Sigma_\epsilon \end{array}$, and for each ε, a global attractor å_ε.

Definition 11.

We say that a family {K_ε}_ε∈[01] is upper semicontinuous at 0 in a metric space (X, d) if given sequences {ε_n}_n∈ℕ ⊂ (0,1] and x_n ∈ K_{ε_n}, with ε_n ∈ 0+ as n → ∞, there exists a convergent subsequence of {x_n}_{n ∈ ℕ} with limit belonging to the closure of K₀ in (X, d).

The previous definition is equivalent to the following: a family {K_ε }_ε∈[0,1] is upper semicontinuous at 0 in a metric space (X, d) if

$\lim_{ϵ \to 0^{+}} d_{H} (K_{ϵ}, K_{0}) = 0.$ $$\begin{array}{} \displaystyle \lim_{\epsilon \to 0^+}{\rm d}_H(K_\epsilon,K_0)=0. \end{array}$$

To prove the upper semicontinuity of ${A_{ϵ}}_{ϵ \in [0, 1]}$ $\begin{array}{} \displaystyle \{\mathbb{A}_\epsilon\}_{\epsilon\in [0,1]} \end{array}$, we set the base space as $H_{0}^{1} (Ω) \times L_{*}$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \times {{\scr L}_ * } \end{array}$, with a metric ∂ defined by

$d [(u_{1}, H_{1}), (u_{2}, H_{2})] = ‖ u_{1} - u_{2} ‖_{H_{0}^{1} (Ω)} + d_{*} (H_{1}, H_{2}),$ $$\begin{array}{} \displaystyle \mathfrak{d}[(u_1,H_1),(u_2,H_2)]=\|u_1-u_2\|_{H^1_0(\Omega)}+d_\ast(H_1,H_2), \end{array}$$(22)

for all $(u_{1}, H_{1}), (u_{2}, H_{2}) \in H_{0}^{1} (Ω) \times L_{*}$ $\begin{array}{} \displaystyle ({u_1},{H_1}),({u_2},{H_2}) \in H_0^1(\Omega ) \times {{\scr L}_ * } \end{array}$.

Before studying the upper semicontinuity of the family of attractors {å_ε}_ε∈[0,1], we will need some convergence results.

Lemma 22.

If(H4)holds and h_ε ∈ 𝒢_ε, we have

$\sup_{t \in ℝ} | h_{ϵ} (t, s) - f (s) | ⩽ β (ϵ) (1 + | s |^{ρ - 1}),$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}|h_\epsilon(t,s)-f(s)|\leqslant \beta(\epsilon)(1+|s|^{\rho-1}), \end{array}$$

for all s ∈ ℝ.

Proof. Let h_ε ∈ 𝒢_ε and (t_m} be a real sequence such that

$\lim_{m \to \infty} d_{2} (h_{ϵ}, θ_{t_{m}} g_{ϵ}) = \lim_{m \to \infty} \sup_{t, s \in ℝ} | h_{ϵ} (t, s) - θ_{t_{m}} g_{ϵ} (t, s) | = 0,$ $$\begin{array}{} \displaystyle \lim_{m\to \infty}d_2(h_\epsilon,\theta_{t_m}g_\epsilon)=\lim_{m\to \infty}\sup_{t,s\in \mathbb{R}}|h_\epsilon(t,s)-\theta_{t_m}g_\epsilon(t,s)|=0, \end{array}$$

where d₂ is as in Subsection 3.1. Hence

$\begin{matrix} sup_{t \in R} | h_{ϵ} (t, s) & - f (s) | ⩽ sup_{t \in R} | h_{ϵ} (t, s) - g_{ϵ} (t + t_{m}, s) | + sup_{t \in R} | g_{ϵ} (t + t_{m}, s) - f (s) | \\ ⩽ d_{2} (h_{ϵ}, θ_{t_{m}} g_{ϵ}) + sup_{t \in R} | g_{ϵ} (t, s) - f (s) | \\ ⩽ d_{2} (h_{ϵ}, θ_{t_{m}} g_{ϵ}) + β (ϵ) (1 + | s |^{ρ - 1}), \end{matrix}$ $$\begin{equation} \begin{split} \sup\limits_{t\in \mathbb{R}}|h_\epsilon(t,s)&-f(s)|\leqslant \sup\limits_{t\in \mathbb{R}}|h_\epsilon(t,s)-g_{\epsilon}(t+t_m,s)|+\sup\limits_{t\in \mathbb{R}}|g_{\epsilon}(t+t_m,s)-f(s)| \\&\leqslant d_2(h_\epsilon,\theta_{t_m}g_\epsilon)+\sup\limits_{t\in \mathbb{R}}|g_{\epsilon}(t,s)-f(s)|\\& \leqslant d_2(h_\epsilon,\theta_{t_m}g_\epsilon) + \beta(\epsilon)(1+|s|^{\rho-1}), \end{split} \end{equation}$$

and making m → ∞ we obtain the result.

Lemma 23.

Assume that(H4)and(C)hold. Then there exists a constant c > 0 such that

$\sup_{h_{ϵ} \in G_{ϵ}} \sup_{t \in ℝ} ‖ h_{ϵ}^{e} (t, u) - f^{e} (u) ‖_{L^{\frac{2 n}{n + 2}} (Ω)} ⩽ c β (ϵ) (1 + ‖ u ‖_{H_{0}^{1} (Ω)}^{ρ - 1}),$ $$\begin{array}{} \displaystyle \sup_{h_\epsilon\in \mathscr{G}_\epsilon}\sup_{t\in \mathbb{R}}\|h_\epsilon^e(t,u)-f^e(u)\|_{L^{\frac{2n}{n+2}}(\Omega)}\leqslant c\beta(\epsilon)(1+\|u\|^{\rho-1}_{H^1_0(\Omega)}), \end{array}$$

for all $u \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle u \in H_0^1(\Omega ) \end{array}$.

Proof. Proceeding as in Lemma 7 and using Lemma 22 we have

$\begin{matrix} sup_{t \in R} ∥ h_{ϵ}^{e} (t, u) & - f^{e} (u) ∥_{L^{\frac{2 n}{n + 2}} (Ω)} = {(\int_{Ω} | h_{ϵ}^{e} (t, u) - f^{e} (u) |^{\frac{2 n}{n + 2}})}^{\frac{n + 2}{2 n}} \\ ⩽ β (ϵ) {(\int_{Ω} (1 + | u |^{ρ - 1})^{\frac{2 n}{n + 2}})}^{\frac{n + 2}{2 n}} ⩽ \tilde{c} β (ϵ) (1 + ∥ u ∥_{L^{\frac{n (ρ - 1)}{2}} (Ω)}^{ρ - 1}) \\ ⩽ c β (ϵ) (1 + ∥ u ∥_{H_{0}^{1} (Ω)}^{ρ - 1}), \end{matrix}$ $$\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\|h_\epsilon^e(t,u)&-f^e(u)\|_{L^{\frac{2n}{n+2}}(\Omega)}=\left[\int_\Omega |h^e_\epsilon(t,u)-f^e(u)|^{\frac{2n}{n+2}}\right]^{\frac{n+2}{2n}}\\ &\leqslant \beta(\epsilon)\left[\int_\Omega(1+|u|^{\rho-1})^{\frac{2n}{n+2}}\right]^{\frac{n+2}{2n}}\leqslant \tilde{c}\beta(\epsilon)(1+\|u\|^{\rho-1}_{L^{\frac{n(\rho-1)}{2}}(\Omega)})\\ &\leqslant c\beta(\epsilon)(1+\|u\|^{\rho-1}_{H^1_0(\Omega)}), \end{array}$$

and the result follows.

Corollary 24.

If(H4)and(C)hold, we have that there exists a constant $\hat{C} > 0$ $\begin{array}{} \displaystyle \hat C > 0 \end{array}$such that

$\sup_{h_{ϵ} \in G_{ϵ}} d_{2}^{e} (h_{ϵ}^{e}, f^{e}) ⩽ \hat{C} β (ϵ) .$ $$\begin{array}{} \displaystyle \sup_{h_\epsilon\in \mathscr{G}_\epsilon}d^e_2(h^e_\epsilon,f^e)\leqslant \hat{C}\beta(\epsilon). \end{array}$$

Proposition 25.

Assume that(H4)and(C)hold true and consider the family (Σ_ε }_ε∈[01]given in Definition 10. Then we have that given sequences (ε_n}_{n ∈ ℕ} ⊂ (0,1] with ε_n → 0+ and H_n ∈ Σ_{ε_n}, for each n ∈ ℕ, there exists a convergent subsequence of (H_n}_{n ∈ ℕ} in (𝒞*, d*), with its limit belonging to Σ₀.

Proof. Since H_n ∈ , item (b) of Lemma 12 implies that there exists λ_n ∈ Γ_{ε_n} and h_n ∈ 𝒢_εn such that

Hn = B_Xnh^en - A_Xn, for each n e N.

$H_{n} = B_{λ_{n}} h_{n}^{e} - {\tilde{A}}_{λ_{n}}, for each n \in ℕ .$ $$\begin{array}{} \displaystyle H_n = B_{\lambda_n}h_n^e-\tilde{A}_{\lambda_n}, \hbox{ for each } n\in \mathbb{N}. \end{array}$$

Since Γ is compact (recall Remark 7), there exists a subsequence (λ_nk} that converges to a function λ₀ in (𝒞₁,d₁). Now, Corollary 24 shows that $d_{2}^{e} (h_{n}^{e}, f^{e}) ⩽ \hat{C} β (ϵ_{n}) \to 0$ $\begin{array}{} \displaystyle d_2^e(h_n^e,{f^e}) \leqslant \hat C\beta ({\epsilon_n}) \to 0 \end{array}$ as n → ∞ and hence $h_{n}^{e}$ $\begin{array}{} \displaystyle h_n^e \end{array}$ converges to f^e in $(C_{2}^{e}, d_{2}^{e})$ $\begin{array}{} \displaystyle ({\scr C}_2^e,d_2^e) \end{array}$. Therefore, we can easily see that H_nk converges to $B_{λ_{0}} f^{e} - {\tilde{A}}_{λ_{0}}$ $\begin{array}{} \displaystyle B_{\lambda_0}f^e-\tilde{A}_{\lambda_0} \end{array}$ in (𝒞*, d*), which is in Σ₀ by item (a) of Lemma 12.

With these preliminaries results, we are able to begin the proof of the upper semicontinuity of the family of the global attractors (å_ε}_ε∈[01] of the skew-product semiflows, at ε = 0.

Lemma 26.

If {(u_ε,H_ε)}_{ε ∈(0,1]} is such that $(u_{ϵ}, H_{ϵ}) \subset X_{ϵ}$ $\begin{array}{} \displaystyle (u_\epsilon,H_\epsilon)\subset \mathbb{X}_\epsilon \end{array}$ and there exists $(u_{0}, H_{0}) \in X_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{X}_0 \end{array}$ such that $d [(u_{ϵ}, H_{ϵ}), (u_{0}, H_{0})] \to 0$ $\begin{array}{} \displaystyle \mathfrak{d}[(u_\epsilon,H_\epsilon),(u_0,H_0)]\to 0 \end{array}$ as ε → 0⁺, we have

$\begin{matrix} d [Π_{ϵ} (t) (u_{ϵ}, H_{ϵ}), Π_{0} (t) (u_{0}, H_{0})] \overset{ϵ \to 0^{+}}{⟶} 0, f o r e a c h t ⩾ 0. \end{matrix}$ $$\begin{array}{} \displaystyle \mathfrak{d}[\Pi_\epsilon(t)(u_\epsilon,H_\epsilon),\Pi_0(t)(u_0,H_0)]\stackrel{\epsilon\to 0^+}{\longrightarrow} 0, \;{ for \; each }\; t \geqslant 0. \end{array}$$

Proof. We can write

$\begin{matrix} Π (t) (u_{ϵ}, H_{ϵ}) = (φ_{ϵ} (t, H_{ϵ}) u_{ϵ}, θ_{t} H_{ϵ}), for each ϵ \in [0, 1] . \end{matrix}$ $$\begin{array}{} \displaystyle \Pi(t)(u_\epsilon,H_\epsilon)=(\varphi_\epsilon(t,H_\epsilon)u_\epsilon,\theta_tH_\epsilon), \hbox{ for each } \epsilon \in [0,1]. \end{array}$$

Since d_*(H_ε,H₀) → 0 and {θ: t⩾0} is continuous in 𝒞_* for each t ⩾ 0, we easily obtain that d_*(θ_tH_ε,θ_tH₀) → 0, as ε → 0⁺.

It remains to show that $\begin{matrix} ∥ φ_{ϵ} (t, H_{ϵ}) u_{ϵ} - φ_{0} (t, H_{0}) u_{0} ∥_{H_{0}^{1} (Ω)} \to 0 \end{matrix}$ $\begin{array}{} \|\varphi_\epsilon(t,H_\epsilon)u_\epsilon-\varphi_0(t,H_0)u_0\|_{H^1_0(\Omega)}\to 0 \end{array}$, as ε → 0⁺. Using Lemma 18, we can write

$\begin{matrix} φ_{ϵ} (t, H_{ϵ}) u_{ϵ} - φ_{0} (t, H_{0}) u_{0} \\ = φ λ_{ϵ} (t) u_{ϵ} - φ λ_{0} (t) u_{0} + \int_{0}^{t} [B_{λ_{ϵ}} (S) h_{ϵ}^{e} (s, φ_{ϵ} (s, H_{ϵ}) u_{ϵ}) - B_{λ_{0}} (S) f^{e} (φ_{0} (s, H_{0}) u_{0})] d s, \end{matrix}$ $$\begin{array}{} \displaystyle \varphi_\epsilon&(t,H_\epsilon)u_\epsilon-\varphi_0(t,H_0)u_0 \\ &= \varphi_{\lambda_\epsilon}(t)u_\epsilon-\varphi_{\lambda_0}(t)u_0+\int_0^t [B_{\lambda_\epsilon}(s)h_\epsilon^e(s,\varphi_\epsilon(s,H_\epsilon)u_\epsilon)-B_{\lambda_0}(s)f^e(\varphi_0(s,H_0)u_0)]ds, \end{array}$$

where we assumed $H_{ϵ} = B_{λ_{ϵ}} h_{ϵ}^{e} - {\tilde{A}}_{λ_{ϵ}}$ $\begin{array}{} \displaystyle H_\epsilon=B_{\lambda_\epsilon}h_\epsilon^e-\tilde{A}_{\lambda_\epsilon} \end{array}$ and $H_{0} = B_{λ_{0}} f^{e} - {\tilde{A}}_{λ_{0}}$ $\begin{array}{} \displaystyle H_0=B_{\lambda_0}f^e-\tilde{A}_{\lambda_0} \end{array}$.

We have, using Lemma 17, we obtain

$\begin{matrix} ‖ φ_{λ_{ϵ}} (t) u_{ϵ} - φ_{λ_{0}} (t) u_{0} ‖_{H_{0}^{1} (Ω)} ⩽ ‖ φ_{λ_{ϵ}} (t) u_{ϵ} - φ_{λ_{ϵ}} (t) u_{0} ‖_{H_{0}^{1} (Ω)} + ‖ φ_{λ_{ϵ}} (t) u_{0} - φ_{λ_{0}} (t) u_{0} ‖_{H_{0}^{1} (Ω)} \\ ⩽ K ‖ u_{ϵ} - u_{0} ‖_{H_{0}^{1} (Ω)} e^{- k t} + ‖ φ_{λ_{ϵ}} (t) u_{0} - φ_{λ_{0}} (t) u_{0} ‖ H_{0}^{1} (Ω) . \end{matrix}$ $$\begin{array}{} \displaystyle \|\varphi_{\lambda_\epsilon}(t)u_\epsilon&-\varphi_{\lambda_0}(t)u_0\|_{H^1_0(\Omega)} \leqslant \|\varphi_{\lambda_\epsilon}(t)u_\epsilon-\varphi_{\lambda_\epsilon}(t)u_0\|_{H^1_0(\Omega)}+\|\varphi_{\lambda_\epsilon}(t)u_0-\varphi_{\lambda_0}(t)u_0\|_{H^1_0(\Omega)}\\ &\leqslant K\|u_\epsilon-u_0\|_{H^1_0(\Omega)}e^{-kt}+\|\varphi_{\lambda_\epsilon}(t)u_0-\varphi_{\lambda_0}(t)u_0\|_{H^1_0(\Omega)}. \end{array}$$

Again, using Lemma 17, item 2 of Remark 7 and the Gronwall inequality, we obtain that

$‖ φ_{λ_{ϵ}} (t) u_{ϵ} - φ_{λ_{0}} (t) u_{0} ‖_{H_{0}^{1} (Ω)} = O (ϵ) .$ $$\begin{array}{} \displaystyle \|\varphi_{\lambda_\epsilon}(t)u_\epsilon-\varphi_{\lambda_0}(t)u_0\|_{H^1_0(\Omega)} ={O}(\epsilon). \end{array}$$

For the second term, we have

$\begin{matrix} ∥ B_{λ_{ϵ}} (s) h_{ϵ}^{e} (s, φ_{ϵ} (s, H_{ϵ}) u_{ϵ}) & - B_{λ_{0}} (s) f^{e} (φ_{0} (s, H_{0}) u_{0}) ∥_{H_{0}^{1} (Ω)} \\ ⩽ ∥ B_{λ_{ϵ}} (s) [h_{ϵ}^{e} (s, φ_{ϵ} (s, H_{ϵ}) u_{ϵ}) - f^{e} (φ_{ϵ} (s, H_{ϵ}) u_{ϵ})] ∥_{H_{0}^{1} (Ω)} \\ + ∥ [B_{λ_{ϵ}} (s) - B_{λ_{0}} (s)] f^{e} (φ_{0} (s, H_{0}) u_{0}) ∥_{H_{0}^{1} (Ω)} \end{matrix}$ $$\begin{equation} \begin{split} \|B_{\lambda_\epsilon}(s)h_\epsilon^e(s,\varphi_\epsilon(s,H_\epsilon)u_\epsilon)&-B_{\lambda_0}(s)f^e(\varphi_0(s,H_0)u_0)\|_{H^1_0(\Omega)}\\ & \leqslant \|B_{\lambda_\epsilon}(s)[h_\epsilon^e(s,\varphi_\epsilon(s,H_\epsilon)u_\epsilon) - f^e(\varphi_\epsilon(s,H_\epsilon)u_\epsilon)]\|_{H^1_0(\Omega)} \\&+\|[B_{\lambda_\epsilon}(s)-B_{\lambda_0}(s)]f^e(\varphi_0(s,H_0)u_0)\|_{H^1_0(\Omega)} \end{split} \end{equation}$$

and hence, using again item 2 of Remark 7 we obtain that

$\begin{matrix} \int_{0}^{t} ‖ B_{λ_{ϵ}} (s) h_{ϵ}^{e} (s, φ_{ϵ} (s, H_{ϵ}) u_{ϵ}) & - B_{λ_{0}} (s) f^{e} (φ_{0} (s, H_{0}) u_{0}) ‖_{H_{0}^{1} (Ω)} \\ ⩽ O (ϵ) + \int_{0}^{t} ‖ φ_{ϵ} (s, H_{ϵ}) u_{ϵ} - φ_{0} (s, H_{0}) u_{0} ‖ H_{0}^{1} (Ω) d s . \end{matrix}$ $$\begin{array}{} \displaystyle \int_0^t\|B_{\lambda_\epsilon}(s)h_\epsilon^e(s,\varphi_\epsilon(s,H_\epsilon)u_\epsilon)&-B_{\lambda_0}(s)f^e(\varphi_0(s,H_0)u_0)\|_{H^1_0(\Omega)}\\ & \leqslant {O}(\epsilon)+\int_0^t \|\varphi_\epsilon(s,H_\epsilon)u_\epsilon-\varphi_0(s,H_0)u_0\|_{H^1_0(\Omega)}ds. \end{array}$$

Finally, joining the estimates and applying again the Gronwall inequality, we obtain that

$‖ φ_{ϵ} (t, H_{ϵ}) u_{ϵ} - φ_{0} (t, H_{0}) u_{0} ‖_{H_{0}^{1} (Ω)} ⩽ O (ϵ),$ $$\begin{array}{} \displaystyle \|\varphi_\epsilon(t,H_\epsilon)u_\epsilon-\varphi_0(t,H_0)u_0\|_{H^1_0(\Omega)}\leqslant {O}(\epsilon), \end{array}$$

and concludes the result.

We can prove the following:

Lemma 27.

If {(u_ε,H_ε)}_ε∈(0,1]is such that $(u_{ϵ}, H_{ϵ}) \in A_{ϵ}$ $\begin{array}{} \displaystyle (u_\epsilon,H_\epsilon)\in \mathbb{A}_\epsilon \end{array}$and

$\lim_{ϵ \to 0^{+}} d [(u_{ϵ}, H_{ϵ}), (u_{0}, H_{0})] = 0$ $$\begin{array}{} \displaystyle \lim_{\epsilon\to 0^+}\mathfrak{d}[(u_\epsilon,H_\epsilon),(u_0,H_0)]= 0 \end{array}$$

for some $(u_{0}, H_{0}) \in H_{0}^{1} (Ω) \times C_{*}$ $\begin{array}{} \displaystyle (u_0,H_0)\in H^1_0(\Omega)\times \mathscr{C}_\ast \end{array}$, then $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$.

Proof. Using the characterization of global attractors in Subsection 2.1, we know that through each (u_ε, H_ε) ∈ 𝔸_ε we have a global bounded solution $ξ_{ϵ} : ℝ \to X_{ϵ}$ $\begin{array}{} \displaystyle \xi_\epsilon\colon \mathbb{R}\to \mathbb{X}_\epsilon \end{array}$ of {∏_ε(t) : t ⩾ 0}. To show that $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$, it is sufficient to prove that through (u₀, H₀) there exists a global bounded solution $ξ_{0} : ℝ \to X_{0}$ $\begin{array}{} \displaystyle \xi_0\colon \mathbb{R}\to \mathbb{X}_0 \end{array}$ of {∏_ε(t) : t ⩾ 0}.

For any t ⩾ 0, we define ξ₀(t) = ∏(t)(u₀,H₀). Since {Π₀(t):t ⩾ 0} has a global attractor, the set ξ₀([0,∞)) is bounded.

We now use an induction argument to define the solution for negative values of t. Consider the family {ξ_ε(—1)}_ε∈[0,1], which we can write as

$ξ_{ϵ} (- 1) = (u_{ϵ} (- 1), θ_{- 1} H_{ϵ}) .$ $$\begin{array}{} \displaystyle \xi_\epsilon(-1)=(u_\epsilon(-1),\theta_{-1}H_\epsilon). \end{array}$$

Using (21), we have that the family {u_ε(—1)}_ε∈[0,1] and hence there exists a sequence ε_1,n → 0⁺ and a point $u_{- 1} \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle {u_{ - 1}} \in H_0^1(\Omega ) \end{array}$ such that

$u_{ϵ_{1, n}} (- 1) \to u_{- 1}, in H_{0}^{1} (Ω) .$ $$\begin{array}{} \displaystyle u_{\epsilon_{1,n}}(-1)\to u_{-1}, \hbox{ in } H^1_0(\Omega). \end{array}$$

We define then ξ₀(—1) = (u_—1, θ_—1H₀) and ξ(t) = ∏₀(t + 1)(u_—1,θ_—1H₀), for —1 ⩽ t < 0. Clearly, using Lemma 26, we have

$(u_{ϵ_{1, n}}, H_{ϵ_{1, n}}) = Π_{ϵ_{1, n}} (1) ξ_{ϵ_{1, n}} (- 1) \to Π_{0} (1) ξ_{0} (- 1),$ $$\begin{array}{} \displaystyle (u_{\epsilon_{1,n}},H_{\epsilon_{1,n}})=\Pi_{\epsilon_{1,n}}(1)\xi_{\epsilon_{1,n}}(-1)\to \Pi_0(1)\xi_0(-1), \end{array}$$

and thus (u₀,H₀) = ξ₀(0) = ∏₀(1)ξ₀(—1).

Proceeding inductively, for each k ∈ ℕ, we obtain a subsequence {ε_k,n}_n∈ℕ of {ε_k−1,n}_n∈ℕ with ε_k,n → 0⁺ as n → ⊂ and a point $u_{- k} \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle u_{-k}\in H^1_0(\Omega) \end{array}$ such that if ξ_ε(—k) = (u_ε(—k), θ_—kH_ε) we have

$u_{ϵ_{k, n}} (- j) \to u_{- j}, for all j = 1, \dots, k .$ $$\begin{array}{} \displaystyle u_{\epsilon_{k,n}}(-j)\to u_{-j}, \hbox{ for all } j=1,\cdots, k. \end{array}$$

Defining ξ₀(—k) = (u_—k, θ_—kH₀) and ξ₀(t) = ∏₀(t + k)ξ₀(—k) for —k ⩽ t < —k + 1, we have that

$(u_{ϵ_{k, n}}, H_{ϵ_{k, n}}) = Π_{ϵ_{k, n}} (1) ξ_{ϵ_{k, n}} (- j) \to Π_{0} (- j + 1) ξ_{0} (- j + 1), for each j = 1 \dots, k,$ $$\begin{array}{} \displaystyle (u_{\epsilon_{k,n}},H_{\epsilon_{k,n}})=\Pi_{\epsilon_{k,n}}(1)\xi_{\epsilon_{k,n}}(-j)\to \Pi_0(-j+1)\xi_0(-j+1), \hbox{ for each } j=1\cdots,k, \end{array}$$

and thus (u₀,H₀) = ξ₀(0) = ∏₀(k)ξ₀(—k).

Therefore, we obtain that $ξ_{0} : ℝ \to X_{0}$ $\begin{array}{} \displaystyle \xi_0\colon \mathbb{R}\to \mathbb{X}_0 \end{array}$ is a bounded global solution of {∏₀(t): t ⩾ 0} through (u₀,H₀), which implies that $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$ and concludes the proof.

Now, we can easily prove the upper semicontinuity of the family of global attractors {𝔸_ε}_ε∈[0,1].

Theorem 28.

The family of global attractors {𝔸_ε}_ε∈[0,1]is upper semicontinuous at 0.

Proof. If {ε_n, H_n)}_n∈ℕ ⊂ (0, 1] with ε_n → 0 and (u_n, H_n) ∈ 𝔸_{ε_n}, it is clear that there exists a convergent subsequence of {(u_n, H_n)}_n∈ℕ to a point $(u_{0}, H_{0}) \in X_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{X}_0 \end{array}$ (using (21)). Hence, Lemma 27 shows that $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$, which concludes the proof.

6.1

Upper semicontinuity for other attractors

As in immediate consequence of Theorem 28 we have (recall Theorem 21):

Corollary 29.

Assume that(H1)-(H4)and(C)hold true. Then we have that

the family of uniform attractors {𝒜_ε }_ε∈[0,1];

the family of cocycle attractors {A(H_ε)}_{H_ε∈Σ_ε}and

the family of pullback attractors {A_{H_ε}(t)}_t∈ℝ

are upper semicontinuous at 0 in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H^1_0(\Omega) \end{array}$.

Topological structure of attractors

Following the results of [7, Section 3], we will study the structure of the global attractors 𝔸_ε for the skew-product semiflows (∏_ε(t): t ⩾ 0} and using this structure we obtain informations about the structure for the other attractors defined in Theorem 21.

7.1

Structure of 𝔸₀

To study the structure of the global attractors 𝔸_ε, we will study in more detail the structure of the global attractor 𝔸₀ and we will make the following assumption:

There exists a finite number of isolated equilibia ℰ = {e₁, … ,e_p} of

${\begin{matrix} - Δ u = f (u), & in Ω \\ u = 0, & on \partial Ω . \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} { - {\rm{\Delta }}u = f(u),}&{{\rm{in\;\Omega }}}\\ {\quad \;\:{\rm{n\& }}\quad u = 0,}&{{\rm{on\;}}\partial {\rm{\Omega }}.} \end{array}\right. \end{array}$$(F)

With this assumption, we define

$E_{i} = {e_{i}} \times Σ_{0} \subset X_{0}, for i = 1, \dots, p .$ $$\begin{array}{} \displaystyle \mathbb{E}_i=\{e_i\}\times \Sigma_0\subset \mathbb{X}_0, \hbox{ for } i=1,\cdots,p. \end{array}$$

Lemma 30.

Each set $E_{i}$ $\begin{array}{} \displaystyle \mathbb{E}_i \end{array}$, i = 1, …, p, is invariant by the skew-product semiflow (∏₀(t): t ⩾ 0} and $E_{i} \subset A_{0}$ $\begin{array}{} \displaystyle \mathbb{E}_i\subset \mathbb{A}_0 \end{array}$, for each i = 1, …, p.

Proof. Clearly, if we take H₀ ∈ Σ₀, the solution [0, ∞) ∋ ↦ φ(t,H₀)e_i of (10) is the constant solution φ₀(t,H₀)e_i = e_i, for all t ⩾ 0; hence

$Π_{0} (t) (e_{i}, H_{0}) = (e_{i}, θ_{t} H_{0}) \in E_{i}, for all t ⩾ 0.$ $$\begin{array}{} \displaystyle {\Pi _0}(t)({e_i},{H_0}) = ({e_i},{\theta _t}{H_0}) \in {\mathbb{E}_i},{\rm{\;for\;all\;}}t \geqslant 0. \end{array}$$

Conversely, if t ⩾ 0 and H ∈ Σ₀ are given, set H₀ = θ_—tH. We have

$Π_{0} (t) (e_{i}, H) = (e_{i}, θ_{t} H) = (e_{i}, H_{0}),$ $$\begin{array}{} \displaystyle {\Pi _0}(t)({e_i},H) = ({e_i},{\theta _t}H) = ({e_i},{H_0}), \end{array}$$

therefore $(e_{i}, H_{0}) \in Π_{0} (t) E_{i}$ $\begin{array}{} \displaystyle ({e_i},{H_0}) \in {\Pi _0}(t){\mathbb{E}_i} \end{array}$. The last claim follows since $E_{i}$ $\begin{array}{} \displaystyle {{\Bbb E}_i} \end{array}$ is abounded invariant subset of $X_{0}$ $\begin{array}{} \displaystyle {{\Bbb X}_0} \end{array}$.

We can now define a functional on $X_{0}$ $\begin{array}{} \displaystyle {{\Bbb X}_0} \end{array}$, which will help us understand the intern structure of 𝔸₀.

Definition 12.

Define the functional $V : X_{0} \to ℝ$ $\begin{array}{} \displaystyle V:{\mathbb{X}_0} \to \mathbb{R} \end{array}$ by

$V (v, H_{0}) = \frac{1}{2} ‖ v ‖_{H_{0}^{1} (Ω)}^{2} - \int_{Ω} W (v),$ $$\begin{array}{} \displaystyle V(v,{H_0}) = \frac{1}{2}v_{H_0^1(\Omega )}^2 - {\smallint _\Omega }W(v), \end{array}$$(23)

where $W (r) = \int_{0}^{r} f (θ) d θ$ $\begin{array}{} \displaystyle W(r) = \smallint _0^rf(\theta )d\theta \end{array}$, for each $(v, H_{0}) \in X_{0}$ $\begin{array}{} \displaystyle (v,{H_0}) \in {\mathbb{X}_0} \end{array}$.

Lemma 31.

Let (∏₀(t): t ⩾ 0} be the skew-product semiflow defined in (17) for ε = 0. If $(u_{0}, H_{0}) \in X_{0}$ $\begin{array}{} \displaystyle ({u_0},{H_0}) \in {\mathbb{X}_0} \end{array}$and V is the functional in defined in (23), we have that

(a)the map [0, ∞) ∋ t ↦ V(∏₀(t)(u₀,H₀)) is non-increasing, for each $(u_{0}, H_{0}) \in X_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{X}_0 \end{array}$and it is constant in $E_{i}$ $\begin{array}{} \displaystyle {{\Bbb E}_i} \end{array}$, i = 1,… ,p.

(b)If the map [0, ∞) ∋ t ↦ V(∏₀(t)(u₀,H₀)) is constant, then $(u_{0}, H_{0}) \in E_{i}$ $\begin{array}{} \displaystyle ({u_0},{H_0}) \in {{\Bbb E}_i} \end{array}$, for some i = 1, …, p.

Proof. Since V(∏₀(t)(u₀, H₀)) = V(φ₀(t)(u₀, H₀)), it is a straightforward computation to see, if H₀ = B_λf^e - Ãλ for some λ∈Г, that

$\frac{d}{d t} V (Π_{0} (t) (u_{0}, H_{0})) = - {‖ \frac{d}{d t} φ (t, H_{0}) u_{0} ‖}_{L^{2} (Ω)}^{2} - λ (t) {‖ \frac{d}{d t} φ (t, H_{0}) u_{0} ‖}_{H_{0}^{1} (Ω)}^{2} ⩽ 0,$ $$ \frac{d}{dt}V(\Pi_0(t)(u_0,H_0))=-\left\|\frac{d}{dt}\varphi(t,H_0)u_0\right\|^2_{L^2(\Omega)}-\lambda(t)\left\|\frac{d}{dt}\varphi(t,H_0)u_0\right\|^2_{H^1_0(\Omega)}\leqslant 0, $$

since 0 < γ ⩽ λ(t) for all t ∈ ℝ so the map [0, ∞) ∋ t ↦ V(∏₀(t)(u₀,H_o)) is non-increasing, and it is clearly constant in each 𝔼_i.

Now, if map [0, ∞) ∋ t ↦ V(∏₀(t)(u₀,H_o)) is constant, we have that φ₀(t,H₀)u₀ = u₀, for all t ⩾ 0, and hence u₀ is a equilibrium of —∆u = f(u) in $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H_0^1(\Omega ) \end{array}$(Ω), which implies that u₀= e_i, for some i = 1, … , p.

Definition 13.

Let $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$. We define the ω-limit of (u₀,H₀) by

$\begin{matrix} ω (u_{0}, H_{0}) = {(u, H) \in A_{0} : there exists a sequence t_{n} \to \infty \\ \begin{matrix} such that Π_{0} (t_{n}) (u_{0}, H_{0}) \overset{n \to \infty}{\to} (u, H)}, \end{matrix} \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{r}} {\omega ({u_0},{H_0}) = \{ (u,H) \in {_0}:{\rm{\;there\;exists\;a\;sequence\;}}{t_n} \to \infty }\\ {\begin{array}{*{20}{l}} {{\rm{\;\;}}\qquad {\rm{such\;that\;}}{\Pi _0}({t_n})({u_0},{H_0})\mathop \to \limits^{n \to \infty } (u,H)\} ,} \end{array}} \end{array} \end{array}$$

and if $ξ : ℝ \to A_{0}$ $\begin{array}{} \displaystyle \xi:\mathbb{R}\to \mathbb{A}_0 \end{array}$ is a global solution of {∏₀(t): ⩾ 0} through (u₀,H₀), we define the αξ -limit of (u₀,H₀) by

$\begin{matrix} α_{ξ} (u_{0}, H_{0}) = {(u, H) \in A_{0} : there exists a sequence t_{n} \to \infty \\ such that ξ (- t_{n}) \overset{n \to \infty}{\to} (u, H)}, \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {{\alpha _\xi }({u_0},{H_0}) = \{ (u,H) \in \,{_0}:{\rm{\;there\;exists\;a\;sequence\;}}{t_n} \to \infty }\\ {{\rm{\;\;}}\qquad \qquad \qquad \qquad {\rm{such\;that\;}}\xi ( - {t_n})\mathop \to \limits^{n \to \infty } (u,H)\} ,} \end{array} \end{array}$$

It is a well known result, since {∏₀(t): t ⩾ 0} has a global attractor $A_{0}$ $\begin{array}{} \displaystyle \mathbb{A}_0 \end{array}$, that both ω(u₀, H₀) and αξ (u₀, H₀) are non-empty, compact, invariant for {∏₀(t): t ⩾ 0} and connected. With these, we can prove the following result.

Lemma 32.

For any $(u_{0}, H_{0}) \in A_{0}$ $\begin{array}{} \displaystyle (u_0,H_0)\in \mathbb{A}_0 \end{array}$and any global solution ξ through (u₀, H₀), there exists i, j = 1, … , p such that

$ω (u_{0}, H_{0}) \subset E_{i} a n d α_{ξ} (u_{0}, H_{0}) \subset E_{j} .$ $$\begin{array}{} \displaystyle \omega(u_0,H_0) \subset \mathbb{E}_i \quad { and } \quad \alpha_\xi(u_0,H_0)\subset \mathbb{E}_j. \end{array}$$

Moreover, if i = j, then u₀ = eu

Proof. Let (u,H) ∈ ω(u₀,H₀) and t_n → ∞ such that ∏₀(t_n)(u₀,H₀) → (u,H). Since V is a continuous functional in $X_{0}$ $\begin{array}{} \displaystyle {{\Bbb X}_0} \end{array}$, we have that V(∏₀(t_n)(u₀,H₀)) → V(u,H), as n — ∞.

Since V(∏₀(‧)(u₀, H₀)) is non-increasing and has a convergent subsequence, we obtain that

$V (Π_{0} (t) (u_{0}, H_{0})) \to V (u, H), as t \to \infty .$ $$\begin{array}{} \displaystyle V({\Pi _0}(t)({u_0},{H_0})) \to V(u,H),{\rm{\;as\;}}t \to \infty . \end{array}$$

V(no(t)(uo,Ho)) → V(u,H), as t → o.

Hence, if (u₁, H₁) is any point in ω(u₀, H₀), we have that V(u₁, H₁) = V(u,H). Since ω(u₀,H₀) is invariant for {∏₀(t): t ⩾ 0} we have that V(∏₀(t)(u,H)) = V(u,H), and then [0,∞) ∋ t → V(∏₀(t)(u,H)) is constant, which implies that $(u, H) \in E_{i}$ $\begin{array}{} \displaystyle (u,H)\in \mathbb{E}_i \end{array}$, for some i = 1, … , p. The connectedness of ω (u₀, H₀) shows us that $ω (u_{0}, H_{0}) \subset E_{i}$ $\begin{array}{} \displaystyle \omega(u_0,H_0)\subset \mathbb{E}_i \end{array}$.

The proof for α_ξ (u0, H0) is analogous and the last assertion is straightforward

Proposition 33.

The family $E = {E_{1}, \dots, E_{p}}$ $\begin{array}{} \displaystyle \mathfrak{E}=\{\mathbb{E}_1,\cdots, \mathbb{E}_p\} \end{array}$ is a disjoint family of isolated invariants for {∏₀(t): t ⩾ 0}; that is, 𝔼_i ⊂ 𝔸₀, Π₍₀₎(t)𝔼_i = 𝔼_i for all t ⩾ 0, there exists δ > 0 such that 𝔼_iis the maximal invariant set for {∏₀(t): t ⩾ 0} in

$O_{δ} (E_{i}) = {(v, H) \in X_{0} : ‖ v - e_{i} ‖_{H_{0}^{1} (Ω)} < δ},$ $$\begin{array}{} \displaystyle \mathscr{O}_{\delta}(\mathbb{E}_i)=\{(v,H)\in \mathbb{X}_0\colon \|v-e_i\|_{H^1_0(\Omega)}<\delta\}, \end{array}$$

for each i = 1, … , p, and also 𝔼_i ∩ 𝔼_j = ∅, if 1 ⩽ i ≠ j ⩽ p.

Proof. It only remains to prove that there exists δ > 0 such that 𝔼_i is the maximal invariant set in $O_{δ} (E_{i})$ $\begin{array}{} \displaystyle {{\scr O}_\delta }({{\Bbb E}_i}) \end{array}$. Let $δ = \frac{1}{2} \min_{1 ⩽ i \neq j ⩽ p} ‖ e_{i} - e_{j} ‖_{H_{0}^{1} (Ω)}$ $\delta = \frac{1}{2}{\min _{1 \leqslant i \ne j \leqslant p}}{e_i} - {e_j}{_{H_0^1(\Omega )}}$.

If 𝔼_i is not the maximal invariant in $O_{δ} (E_{i})$ $\begin{array}{} \displaystyle {{\scr O}_\delta }({_i}) \end{array}$, there exists a global solution ξ of {∏₀(t): t ⩾ 0} such that ξ(ℝ) ⊂ O_δ(𝔼_i), with ξ(ℝ)\𝔼_i ∩ 𝔼_j = ∅. But then the previous lemma shows that ω(ξ(0)) ⊂ 𝔼_i and α_ξ(ξ(0))⊂𝔼_i, which implies that ξ(0) = (e_i,H₀), and therefore ξ(ℝ) ⊂ 𝔼_i, so we reached a contradiction.

All these results combined show us that the semigroup {∏₀(t): t ⩾ 0} is, in fact, a generalized gradient semigroup (see [7,15,26] for more details) with disjoint family of isolated invariants $E = {E_{1}, \dots, E_{p}}$ $\begin{array}{} \displaystyle \mathfrak{E}=\{\mathbb{E}_1,\cdots,\mathbb{E}_p\} \end{array}$ and as a consequence, we can write the global attractor 𝔸₀ as

$A_{0} = \cup_{i = 1}^{p} W^{u} (E_{i}),$ $$ \mathbb{A}_0 = \bigcup_{i=1}^p \mathbb{W}^u(\mathbb{E}_i), $$

where

$\begin{matrix} W^{u} (E_{i}) = {(u, H) & \in A_{0} : there exists a global solution ξ of {Π_{0} (t) : t ⩾ 0} \\ with ξ (0) = (u, H) such that d (ξ (t), E_{i}) \to 0, as t \to - \infty} . \end{matrix}$ $$\begin{equation} \begin{split} \mathbb{W}^u(\mathbb{E}_i)= \{(u,H)&\in \mathbb{A}_0\colon \hbox{ there exists a global solution } \xi \hbox{ of }\{\Pi_0(t)\colon t\geqslant 0\} \\ &\hbox{ with } \xi(0)=(u,H) \hbox{ such that }\mathfrak{d}(\xi(t),\mathbb{E}_i)\to 0, \hbox{ as } t\to -\infty\}. \end{split} \end{equation}$$

If H₀ ∈ Σ₀, there exists λ ∈ Γ such that H₀ = Bλf^e — Ã_λ. Thus we can define

$\begin{matrix} W^{u} (e_{i}, H_{0}) = {u \in & H_{0}^{1} (Ω) : there exists a global solution η (10)for H_{0} \\ with η (0) = u such that ‖ η (t) - e_{i} ‖_{H_{0}^{1} (Ω)} \to 0, as t \to - \infty} . \end{matrix}$ $$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {{W^u}({e_i},{H_0}) = \{ u \in \,{\rm{\& }}H_0^1({\rm{\Omega }}):{\rm{\;there\;exists\;a\;global\;solution\;}}\eta {\rm{\;(10)for \;}}{H_0}}\\ {\qquad \qquad \qquad \qquad \quad {\rm{with\;}}\eta (0) = u{\rm{\;such\;that\;}}\eta (t) - {e_i}{_{H_0^1(\Omega )}} \to 0,{\rm{\;as\;}}t \to - \infty \} .} \end{array} \end{array}$$

It is simple to see that

$W^{u} (E_{i}) = W^{u} (e_{i}, H_{0}) \times {H_{0}}, for each i = 1, \dots, p,$ $$\begin{array}{} \displaystyle {\mathbb{W}^u}({\mathbb{E}_i}) = {W^u}({e_i},{H_0}) \times \left\{ {{H_0}} \right\},{\rm{\;for\;each\;}}i = 1, \cdots ,p, \end{array}$$

and thus we have that

$A_{0} = \cup_{i = 1}^{p} \underset{H_{0} \in Σ_{0}}{\cup} W^{u} (e_{i}, H_{0}) \times {H_{0}},$ $$ \mathbb{A}_0=\bigcup_{i=1}^p\bigcup_{H_0\in \Sigma_0}W^u(e_i,H_0)\times \{H_0\}, $$

and Theorem 21 shows us that the uniform attractor 𝒜₀of the non-autonomous dynamical system ${(φ_{0}, θ)}_{(H_{0}^{1} (Ω), Σ_{0})}$ $\begin{array}{} \displaystyle {({\varphi _0},\theta )_{(H_0^1(\Omega ),{\Sigma _0})}} \end{array}$ is given by

$A_{0} = \cup_{i = 1}^{p} \underset{H_{0} \in Σ_{0}}{\cup} W^{u} (e_{i}, H_{0}) .$ $$ \mathscr{A}_0 = \bigcup_{i=1}^p \bigcup_{H_0\in \Sigma_0} W^u(e_i,H_0). $$

Moreover, the cocycle attractor {A(H₀)}_H₀∈Σ₀ of the non-autonomous dynamical system ${(φ_{0}, θ)}_{(H_{0}^{1} (Ω), Σ_{0})}$ $\begin{array}{} \displaystyle {({\varphi _0},\theta )_{(H_0^1(\Omega ),{\Sigma _0})}} \end{array}$ is given by

$A (H_{0}) = \cup_{i = 1}^{p} W^{u} (e_{i}, H_{0}), for each H_{0} \in Σ_{0} .$ $$ A(H_0) = \bigcup_{i=1}^p W^u(e_i,H_0), \hbox{ for each } H_0\in \Sigma_0. $$

And finally, for each H₀ ∈ Σ₀, the pullback attractor {A_H₀(t)}_t∈ℝ of the evolution process T_H₀(t,s) = φ₀(t — s, θ_sH₀) is given by

$A_{H_{0}} (t) = ⋃_{i = 1}^{p} W^{u} (e_{i}, θ_{t} H_{0}), for all t \in R and each H_{0} \in Σ_{0} .$ $$ A_{H_0}(t) = \bigcup_{i=1}^p W^u(e_i,\theta_tH_0), \hbox{ for all } t\in \mathbb{R} \hbox{ and each } H_0\in \Sigma_0. $$

Remark 11.

If H₀ = B_γf^e — Ã_γ, defining

$A (t) = A_{H_{0}} (t), for each t \in ℝ,$ $$ \mathscr{A}(t)=A_{H_0}(t), \hbox{ for each }t\in \mathbb{R}, $$

we obtain the pullback attractor for (5); that is, for each t ∈ ℝ, we have

$A (t) = {η (t) : η is a bounded global solution of (5)} .$ $$\begin{array}{} \displaystyle \mathscr{A}(t) = \{ \eta (t):\eta {\rm{\;is\;a\;bounded\;global\;solution of (}}5)\} . \end{array}$$

If γ(·) is a constant function, equation (5) is autonomous, Σ₀ = B_γf^e — Ã_γ} is a singleton and we obtain, as a particular case, that the autonomous system generated by (5) is a gradient semigroup with a finite collection of equilibria ℰ= {e₀, …, e_p} and that 𝔸₀ = 𝒜 × Σ₀, where 𝒜₀ = W^u(e_i, H₀) and H₀ = B_γf^e —Ã_γ.

7.2

Structure of 𝔸_e

In this subsection we prove that, under suitable conditions, the attractors 𝔸_ε inherit the same generalized gradient structure from 𝔸₀. To this end, we first need to take the following fact in account: we have, following the ideas of the proof of Lemma 26, if H_ε → H₀ in (𝒞_*,d_*), that

$\lim_{ϵ \to 0^{+}} d [Π_{ϵ} (t) (u, H_{ϵ}), Π_{0} (t) (u, H_{0})] = 0,$ $$\begin{array}{} \displaystyle \lim_{\epsilon \to 0^+}\mathfrak{d}[\Pi_\epsilon(t)(u,H_\epsilon),\Pi_0(t)(u,H_0)]=0, \end{array}$$

uniformly for t in bounded subsets of ℝ, u in bounded subsets of $H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle H^1_0(\Omega) \end{array}$.

With all these considerations and previous results, we are able to state the following structural result.

Theorem 34

Assume that hypotheses (H1)-(H4), (C) and (F) hold true. Assume also that

(a) for each ε ∈ (0,1] there exists a disjoint family of isolated invariants $E_{ϵ} = {E_{1, ϵ}, \dots, E_{p, ϵ}}$ $\begin{array}{} \displaystyle \mathfrak{E}_\epsilon=\{\mathbb{E}_{1,\epsilon},\cdots,\mathbb{E}_{p,\epsilon}\} \end{array}$ for {∏_ε(t): t ⩾ 0} such that

$d_{H} [E_{i, ϵ}, E_{i}] + d_{H} [E_{i}, E_{i, ϵ}] \to 0, as ϵ \to 0^{+};$ $$\begin{array}{} \displaystyle \mathfrak{d}_H[\mathbb{E}_{i,\epsilon},\mathbb{E}_{i}]+\mathfrak{d}_H[\mathbb{E}_{i},\mathbb{E}_{i,\epsilon}]\to 0, \hbox{ as } \epsilon \to 0^+; \end{array}$$

(b) there exists ε₀ > 0 and neighborhoods 𝕌_i of 𝔼_i such that 𝔼_i,εis the maximal invariant set of {∏_ε(t): t ⩾ 0} in 𝕌_i, for each i = 1, …, p and 0 < ε ⩽ ε₀.

Then there exists ε₀ > 0 such that {∏_ε(t): t ⩾ 0} is a generalized gradient semigroup with a disjoint family of isolated invariantsℱ_ε, for each 0 < ε ⩽ ε₁. Moreover for 0 ⩽ ε ⩽ ε₁, we have

$A_{ϵ} = ⋃_{i = 1}^{p} W^{u} (E_{i, ϵ}) .$ $$\mathbb{A}_\epsilon=\bigcup_{i=1}^p\mathbb{W}^u(\mathbb{E}_{i,\epsilon}).$$

Proof. The proof of this theorem is analogous to the proof of [15, Theorem 1.5], with the aid of Proposition 25.

7.3

Global hyperbolic solutions for (10)

Let ξ_ε be a global solution of {∏_ε(t): t ⩾ 0} in 𝔸_ε. Thus we have that there exists H_ε ∈ ∑_ε such that ξ_ε(t) = (η_ε(t), θ_tH_ε), for all t ∈ ℝ, where η_ε is a global bounded solution of (10).

Writing $H_{ϵ} = B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_\epsilon=B_\lambda h_\epsilon^e-\tilde{A}_\lambda \end{array}$, for some λ ∈ Γ and h_ε ∈ 𝒞₂, we can consider the variational problem for the solution η_ε, given by

$(\begin{matrix} u_{t} - λ (t) Δ u_{t} - Δ u = D_{s} h_{ϵ}^{e} (t, η_{ϵ} (t)) u, in Ω \\ u = 0, on \partial Ω, \end{matrix})$ $$\begin{equation} \left\{\begin{split} &u_t-\lambda(t)\Delta u_t-\Delta u=D_sh^e_\epsilon(t,\eta_\epsilon(t))u, \hbox{ in } \Omega\\ &u=0, \hbox{ on } \partial \Omega, \end{split}\right. \end{equation}$$(24)

where $[D_{s} h_{ϵ}^{e} (t, η_{ϵ} (t)) u] (x) = \partial_{s} h_{ϵ} (t, η_{ϵ} (t) (x)) u (x)$ $\begin{array}{} \displaystyle [D_sh_\epsilon^e(t,\eta_\epsilon(t))u](x)=\partial_sh_\epsilon(t,\eta_\epsilon(t)(x))u(x) \end{array}$,a.e x ∈ Ω, for $u \in H_{0}^{1} (Ω)$ $\begin{array}{} \displaystyle u\in H^1_0(\Omega) \end{array}$.

This equation generates a linear evolution process ${L_{ξ_{ϵ}} (t, s) : t ⩾ s} \subset L (H_{0}^{1} (Ω))$ $\begin{array}{} \displaystyle \{L_{\xi_\epsilon}(t,s)\colon t \geqslant s\}\subset \mathscr{L}(H^1_0(\Omega)) \end{array}$, where u(t) = L_{ξ_ε}(t, s)u₀is the solution at time t of (24) with u(s) = u₀.

Definition 14.

We say that L_{ξ_ε}(t, s): t ⩾ s} has an exponential dichotomy if there exists a family of projections {Q(t)}_t∉ℝ in $L (H_{0}^{1} (Ω))$ $\begin{array}{} \displaystyle \mathscr{L}(H^1_0(\Omega)) \end{array}$satisfying:

(a) Q(t)L_{ξ_ε}(t, s) = (t,s)Q(s),forall t ⩾ s.

(b) The restriction L_{ξ_ε}(t,s)|_R(Q(s)), t ⩾ s, is an isomorphism from R(Q(s)) into R(Q(t)); and we denote its inverse by L_{ξ_ε}(s, t): R(Q(t)) → R(Q(s)).

$\begin{matrix} ‖ L_{ξ_{ϵ}} (t, s) (I - Q (s)) ‖_{L (H_{0}^{1} (Ω))} ⩽ M e^{- ω (t - s)}, for t ⩾ s; \\ ‖ L_{ξ_{ϵ}} (s, t) Q (t) ‖_{L (H_{0}^{1} (Ω))} ⩽ M e^{ω (s - t)}, for s < t . \end{matrix}$ $$\begin{array}{} \displaystyle & \|L_{\xi_\epsilon}(t,s)(I-Q(s))\|_{\mathscr{L}(H^1_0(\Omega))}\leqslant Me^{-\omega(t-s)}, \hbox{ for } t \geqslant s;\\ & \|L_{\xi_\epsilon}(s,t)Q(t)\|_{\mathscr{L}(H^1_0(\Omega))}\leqslant Me^{\omega(s-t)}, \hbox{ for } s<t. \end{array}$$

In this case, we say that ξ_ε is a hyperbolic global solution of {Π_ε (t): t ⩾ 0}.

Now we make the following assumption:

$\begin{matrix} Each global & solution ξ_{i, H_{0}}^{*} (t) = (e_{i}, θ_{t} H_{0}), t \in R and \\ H_{0} \in Σ_{0}, is hyperbolic . \end{matrix}$ \begin{equation*} \begin{split} \hbox{ Each global } &\hbox{solution } \xi_{i,H_0}^\ast(t)=(e_i,\theta_tH_0), \ t\in \mathbb{R} \hbox{ and } \\&H_0\in \Sigma_0, \hbox{ is hyperbolic}. \end{split} \end{equation*}(Hy)

Remark 12.

Since $Σ_{0} = {B_{λ} f^{e} - {\tilde{A}}_{λ}}_{λ \in Γ}$ $\begin{array}{} \displaystyle \Sigma_0=\{B_\lambda f^e-\tilde{A}_\lambda\}_{\lambda\in \Gamma} \end{array}$, we have a family {ξ^*_i,λ}_λ∊T of hyperbolic global solutions of {Π₀(t): t ⩾ 0}, where $ξ_{i, λ}^{*} = ξ_{i, H_{0}}^{*}$ $\begin{array}{} \displaystyle \{\xi_{i,\lambda}^\ast\}_{\lambda\in \Gamma} \end{array}$ for $H_{0} = B_{λ} f^{e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_0=B_\lambda f^e-\tilde{A}_\lambda \end{array}$.

Now, proceeding as in [16, Section 7.2] and [26], we have the following result:

Proposition 35.

Assume that hypotheses(H1)-(H5), (C), (F)and(Hy)hold true. Thus, there exist ε₀ > 0 andδ > 0 such that for each i = 1, ··· , p, λ ∊ Γ and ε ∊ (0, ε₀], there exist a unique $h_{ϵ}^{*} \in G_{ϵ}$ $\begin{array}{} \displaystyle h_\epsilon^\ast\in \mathscr{G}_\epsilon \end{array}$and a unique bounded global solution $η_{i, λ, ϵ}^{*} : ℝ \to A_{ϵ}$ $\begin{array}{} \displaystyle \eta_{i,\lambda,\epsilon}^\ast \colon \mathbb{R}\to \mathscr{A}_\epsilon \end{array}$of (10), with $H_{ϵ} = B_{λ} h_{ϵ}^{*, e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_\epsilon=B_\lambda h_\epsilon^{\ast,e}-\tilde{A}_\lambda \end{array}$, which we denote by $H_{ϵ}^{λ}$ $\begin{array}{} \displaystyle H_\epsilon^\lambda \end{array}$, such that:

(i)the function $ℝ ∋ t \mapsto ξ_{i, λ, ϵ}^{*} (t) = (η_{i, λ, ϵ}^{*} (t), θ_{t} H_{ϵ}^{λ}) \in A_{ϵ}$ $\begin{array}{} \displaystyle \mathbb{R}\ni t\mapsto \xi_{i,\lambda,\epsilon}^\ast(t)=(\eta_{i,\lambda,\epsilon}^\ast(t),\theta_tH_\epsilon^\lambda)\in \mathbb{A}_\epsilon \end{array}$is a hyperbolic global solution of {Π_ε (t): t ⩾ 0};

(ii) $\sup_{t \in ℝ} d [ξ_{i, λ, ϵ}^{*} (t), (e_{i}, θ_{t} H_{0}^{λ})] < δ$ $\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\mathfrak{d}[\xi_{i,\lambda,\epsilon}^\ast(t),(e_i,\theta_tH_0^\lambda)]<\delta \end{array}$; where $H_{0}^{λ} = B_{λ} f^{e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_0^\lambda=B_\lambda f^e-\tilde{A}_\lambda \end{array}$, and

(iii) $\sup_{t \in ℝ} d [ξ_{i, λ, ϵ}^{*} (t), (e_{i}, θ_{t} H_{0}^{λ})] \to 0$ $\begin{array}{} \displaystyle \sup_{t\in \mathbb{R}}\mathfrak{d}[\xi_{i,\lambda,\epsilon}^\ast(t),(e_i,\theta_tH_0^\lambda)]\to 0 \end{array}$, asε → 0⁺.

Moreover, ifξ_ε is a global solution of {Π_ε (t): t ⩾ 0} such that $d [ξ_{ϵ} (t), ξ_{i, λ, ϵ}^{*} (t)] < δ$ $\begin{array}{} \displaystyle \mathfrak{d}[\xi_\epsilon(t),\xi_{i,\lambda,\epsilon}^\ast(t)]<\delta \end{array}$for all t ⩾ 0 (t ⩽ 0) then

$d [ξ_{ϵ} (t), ξ_{i, λ, ϵ}^{*} (t)] \to 0 as t \to \infty (t \to - \infty)$ $$\begin{array}{} \displaystyle \mathfrak{d}[\xi_\epsilon(t),\xi_{i,\lambda,\epsilon}^\ast(t)]\to 0 \hbox{ as } t\to \infty \ (t\to -\infty) \end{array}$$

With this proposition, we can construct a family $E_{ϵ} = {E_{1, ϵ}, \dots, E_{p, ϵ}}$ $\begin{array}{} \displaystyle \mathfrak{E}_\epsilon=\{\mathbb{E}_{1,\epsilon},\cdots,\mathbb{E}_{p,\epsilon}\} \end{array}$, satifying the conditions of Theorem 34. In fact, for each λ ∊ Γ, let

$D_{i, ϵ} (λ) = \underset{t \in ℝ}{\cup} ξ_{i, λ, ϵ}^{*} (t) = \underset{t \in ℝ}{\cup} (η_{i, λ, ϵ}^{*} (t), θ_{t} H_{ϵ}^{λ}), for each i = 1, \dots, p,$ $$\begin{array}{} \displaystyle \mathbb{D}_{i,\epsilon}(\lambda)= \bigcup_{t\in \mathbb{R}}\xi_{i,\lambda,\epsilon}^\ast(t)= \bigcup_{t\in \mathbb{R}}(\eta_{i,\lambda,\epsilon}^\ast(t), \theta_tH_\epsilon^\lambda), \hbox{ for each } i=1,\cdots,p, \end{array}$$(25)

and define

$E_{i, ϵ} = \underset{λ \in Γ}{\cup} D_{i, ϵ} (λ), for each i = 1, \dots, p .$ $$\begin{array}{} \displaystyle \mathbb{E}_{i,\epsilon}=\bigcup_{\lambda\in \Gamma}\mathbb{D}_{i,\epsilon}(\lambda), \hbox{ for each } i=1,\cdots,p. \end{array}$$

Now, as an immediate consequence of Theorem 34 and Proposition 35, we have:

Theorem 36.

Assume that hypotheses(H1)-(H5), (C), (F)and(Hy)hold true. Consider the disjoint family of isolated invariants $E_{ϵ} = {E_{1, ϵ}, \dots, E_{p, ϵ}}$ $\begin{array}{} \displaystyle \mathfrak{E}_\epsilon=\{\mathbb{E}_{1,\epsilon},\cdots,\mathbb{E}_{p,\epsilon}\} \end{array}$for {Π_ε(t): t ⩾ 0} given by (25).

Then there exists ε₁ > 0 such that {Π_ε (t): t ⩾ 0} is a generalized gradient semigroup with a disjoint family of isolated invariants ℭ_ε,for each 0 ⩽ ε ⩽ ε₁. Moreover, for 0 ⩽ ε ⩽ ε₁, we have

$A_{ϵ} = \cup_{i = 1}^{p} W^{u} (E_{i, ϵ}),$ $$\begin{array}{} \displaystyle \mathbb{A}_\epsilon=\bigcup_{i=1}^p\mathbb{W}^u(\mathbb{E}_{i,\epsilon}), \end{array}$$

and

$W^{u} (E_{i, ϵ}) = \underset{λ \in Γ}{\cup} W^{u} (D_{i, ϵ} (λ)) .$ $$\begin{array}{} \displaystyle \mathbb{W}^u(\mathbb{E}_{i,\epsilon})=\bigcup_{\lambda\in \Gamma}\mathbb{W}^u(\mathbb{D}_{i,\epsilon}(\lambda)). \end{array}$$

With this result, we can derive structures for the other types of attractors, as we did for 𝔸₀; namely, we have that the uniform attractor of ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$ is given by

$A_{ϵ} = π_{H_{0}^{1} (Ω)} A_{ϵ} = \cup_{i = 1}^{p} W^{u} (E_{i, ϵ}),$ $$\begin{array}{} \displaystyle \mathscr{A}_\epsilon = \pi_{H^1_0(\Omega)}\mathbb{A}_\epsilon = \bigcup_{i=1}^p W^u(\mathscr{E}_{i,\epsilon}), \end{array}$$

where $E_{i, ϵ} = \underset{λ \in Γ}{\cup} \underset{t \in ℝ}{\cup} η_{i, λ, ϵ}^{*} (t)$ $\begin{array}{} \displaystyle \mathscr{E}_{i,\epsilon}=\bigcup_{\lambda\in \Gamma}\bigcup_{t\in \mathbb{R}}\eta_{i,\lambda,\epsilon}^\ast(t) \end{array}$. Also, the cocycle attractor for ${(φ_{ϵ}, θ)}_{(H_{0}^{1} (Ω), Σ_{ϵ})}$ $\begin{array}{} \displaystyle (\varphi_\epsilon,\theta)_{(H^1_0(\Omega),\Sigma_\epsilon)} \end{array}$ is given by

$A (H_{ϵ}) = \cup_{i = 1}^{p} W^{u} (E_{i, ϵ} (λ)),$ $$\begin{array}{} \displaystyle A(H_\epsilon)= \bigcup_{i=1}^p W^u(\mathscr{E}_{i,\epsilon}(\lambda)), \end{array}$$

where $E_{i, ϵ} (λ) = \underset{t \in ℝ}{\cup} η_{i, λ, ϵ}^{*} (t)$ $\begin{array}{} \displaystyle \mathscr{E}_{i,\epsilon}(\lambda)=\bigcup_{t\in \mathbb{R}}\eta_{i,\lambda,\epsilon}^\ast(t) \end{array}$ and $H_{ϵ} = B_{λ} h_{ϵ}^{e} - {\tilde{A}}_{λ}$ $\begin{array}{} \displaystyle H_\epsilon=B_\lambda h_{\epsilon}^e-\tilde{A}_\lambda \end{array}$.

Lastly, if H_ε ∊ Σ_ε, the pullback attractor {A_{H_ε} (t)}_t∊ℝ of the evolution process T_{H_ε} (t, s) = φ_ε (t - s, θ_sH_ε) is given by

$A_{H_{ϵ}} (t) = \cup_{i = 1}^{p} W^{u} (η_{i, λ, ϵ}^{*}) (t), for all t \in ℝ,$ $$\begin{array}{} \displaystyle A_{H_\epsilon}(t) = \bigcup_{i=1}^p W^u(\eta_{i,\lambda,\epsilon}^\ast)(t), \hbox{ for all } t\in \mathbb{R}, \end{array}$$

where

$\begin{matrix} W^{u} (ξ_{i, λ, ϵ}^{*}) (t) & = {u \in H_{0}^{1} (Ω) : there exists a global solution η_{ϵ} of (10) for H_{ϵ} \\ with η_{ϵ} (t) = u such that ∥ η_{ϵ} (s) - η_{i, λ, ϵ}^{*} (s) ∥_{H_{0}^{1} (Ω)} \to 0, as s \to - \infty} . \end{matrix}$ \begin{equation*} \begin{split} W^u(\xi_{i,\lambda,\epsilon}^\ast)(t) &=\{u\in \ H^1_0(\Omega) \colon \hbox{ there exists a global solution } \eta_\epsilon \hbox{ of }(10) \hbox{ for } H_\epsilon \\& \hbox{with }\eta_\epsilon(t)=u \hbox{ such that }\|\eta_\epsilon(s)-\eta_{i,\lambda,\epsilon}^\ast(s)\|_{H^1_0(\Omega)}\to 0, \hbox{ as } s\to -\infty\}. \end{split} \end{equation*}

7.4

Further remarks

As discussed extensively in [5] and [7], we could do the analysis presented in this work, removing condition (C) by assuming only that we have a semigroup of translations {θ_t: t ⩾ 0} in 𝒢_ε with a global attractor Ξ_ε ⊂ 𝒢_ε. Thus, despite some complications with notations, the results would remain unchanged, replacing 𝒢_ε by Ξ_ε. This, in turn, allows us to give a more precise description of the attractors when g_ε is, for instance, asymptotically autonomous; since in this case, we notice that Ξ_ε is a singleton, which is significantly smaller than 𝒢_ε.

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Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

Published Online: Feb 08, 2017

Page range: 31 - 60

Received: Oct 10, 2016

Accepted: Feb 08, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00004

KeywordsNon-autonomous perturbations, non-autonomous dynamical systems, pullback attractors, cocyle attractor, uniform attractor, non-classical parabolic equations, evolution process, skew-product semiflow

© 2017 Matheus C. Bortolan, Felipe Rivero, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Non-autonomous perturbations, non-autonomous dynamical systems, pullback attractors, cocyle attractor, uniform attractor, non-classical parabolic equations, evolution process, skew-product semiflow