On -regularity of global attractors generated by strong solutions of reaction-diffusion equations

This paper is mainly devoted to studying the regularity properties of global attractors for multivalued semiflows generated by strong solutions of reaction-diffusion equations.

The existence and properties of global attractors for dynamical systems generated by reaction-diffusion equations have been studied by many authors over the last thirty years. For equations generating a single-valued semigroup such results are well known since the 80s (see e.g. [5], [6], [7], [8], [24], [34]). Moreover, deep results concerning the structure of the attractors were proved for scalar equations (see [12], [13], [29], [30], [31]). Furthermore, for rather general parabolic equations boundedness of attractors in Sobolev spaces of higher orders were obtained as well (see e.g. [3], [4]).

When uniqueness of the Cauchy problem is not satisfied we have to work with multivalued semiflows rather than semigroups. In this direction, existence and topological properties of global and trajectory attractors have been studied by several authors over the last years (see [2], [9], [10], [11], [14], [16], [20], [22], [23], [25], [26], [36]). However, concerning the structure of the attractor little is known so far in comparison with the single-valued case. Nevertheless, recently the global attractor has been characterised using the unstable manifold of the set of stationary points, generalizing in this way well-known results from the single-valued case (see [17], [18], [19]). In particular, such structure was proved to be true for the global attractor generated by strong solutions of reaction diffusion equations in which the nonlinear term satisfies a critical growth condition.

It is important to point out that there are two different approaches to the study of these equations. One method relies on the construction of weak solutions through Galerkin approximations, whereas the other one makes use of the properties of sectorial operators in order to obtain mild solutions, which are defined by the variation of constants formula. It seems that this has given rise to two separate groups of papers, whose paths have rarely crossed. However, we find it interesting to use the powerful technique of sectorial operators in order to improve the regularity of weak solutions and global attractors in the multivalued case. Indeed, in this paper we prove that under a suitable assumption mild and weak solutions are equivalent, and using this result we are able to improve the regularity of the global attractor generated by strong solutions which was obtained in [17].

This paper is split into three different parts.

In the first section, we prove that the concepts of weak and mild solutions are equivalent provided that an appropriate condition holds.

In the second section, we use the above equivalence in order to show that the global attractor generated by strong solutions of the reaction-diffusion equation is bounded in suitable L^r-spaces in the case where the nonlinear term satisfies a critical growth condition.

Finally, in the third section, considering a supercritical growth condition we define a multivalued semiflow by taking all strong solutions satisfying an energy inequality and then prove that a weak global attractor exists, that is, we construct an attractor which attracts bounded sets of the phase space with respect to the weak topology.

Equivalence of different definitions of solutions

Let Ω ⊂ ℝⁿ, n ≥ 1, be an open bounded subset with sufficiently smooth boundary ∂Ω. We consider the following problem

$\begin{matrix} (\begin{matrix} \frac{\partial u}{\partial t} - Δ u + f (t, u) = h (t, x), x \in Ω, t > τ, \\ u |_{\partial Ω} = 0, \\ u (τ, x) = u_{0} (x), x \in Ω . \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{l} \dfrac{\partial u}{\partial t}-\Delta u+f(t,u)=h(t,x),\quad x\in \Omega,\, t \gt \tau,\\ u|_{\partial\Omega}=0,\\ u(\tau,x)=u_{0}(x)\text{, }x\in\Omega. \end{array} \right. \end{array}$$(1)

The functions f, h are assumed to satisfy the following conditions:

$\begin{matrix} f \in C (R \times R), \end{matrix}$ $$\begin{array}{} \displaystyle f\in\mathbb{C}(\mathbb{R}\times\mathbb{R}), \end{array}$$(2)

$\begin{matrix} h \in L_{l o c}^{2} (R; L^{2} (Ω)), \end{matrix}$ $$\begin{array}{} \displaystyle h\in L_{loc}^{2}(\mathbb{R};L^{2}(\Omega)), \end{array}$$(3)

$\begin{matrix} | f (t, u) | \leq C_{1} (1 + | u |^{p - 1}), \end{matrix}$ $$\begin{array}{} \displaystyle |f(t,u)|\leq C_{1}(1+|u|^{p-1}), \end{array}$$(4)

$\begin{matrix} f (t, u) u \geq α | u |^{p} - C_{2}, \end{matrix}$ $$\begin{array}{} \displaystyle f(t,u)u\geq\alpha|u|^{p}-C_{2}, \end{array}$$(5)

for all (t, u) ∈ ℝ × ℝ, where C₁, C₂ are positive constants, p ≥ 2 and α > 0 if p > 2 but α ∈ ℝ if p = 2.

Denote $\begin{matrix} F (u) = \int_{0}^{u} f (s) d s . \end{matrix}$ $\begin{array}{} F(u)=\int_{0}^{u}f(s)ds. \end{array}$ If p > 2 from (4)-(5) we obtain that $\begin{matrix} \underset{| u | \to \infty}{lim inf} \frac{f (u)}{u} = + \infty \end{matrix}$ $\begin{array}{} \displaystyle \liminf\limits_{|u|\rightarrow\infty}\frac{f(u)}{u}=+\infty \end{array}$ and that there exists D₁, D₂, δ > 0 such that

$\begin{matrix} | F (u) | \leq D_{1} (1 + {(u)}^{p}), F (u) \geq δ {(u)}^{p} - D_{2}, \forall u \in R . \end{matrix}$ $$\begin{array}{} \displaystyle |F(u)|\leq D_{1}(1+\left\vert u\right\vert ^{p}),\,\, F(u)\geq\delta\left\vert u\right\vert ^{p}-D_{2},\quad\forall u\in\mathbb{R}. \end{array}$$(6)

If p = 2, then (6) remains valid but with δ ∈ ℝ.

In the sequel we shall denote by H the space L²(Ω) endowed with the norm ∥⋅∥ and the scalar product (⋅, ⋅), and by V the space $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H_{0}^{1} \end{array}$ (Ω) with the norm ∥⋅∥_V and the scalar product ((⋅, ⋅)), whereas V^′ = H⁻¹(Ω) is the dual space of V with the norm ∥⋅∥_V^′. The pairing between the space V and V^′ will be denoted by 〈⋅, ⋅〉. Also, putting $\begin{matrix} \frac{1}{p} + \frac{1}{q} = 1, \end{matrix}$ $\begin{array}{} \displaystyle \frac{1}{p}+\frac{1}{q}=1, \end{array}$ the pairing between the spaces L^p(Ω) and L^q(Ω) will be given by 〈⋅, ⋅〉_q,p.

Definition 1

The function $\begin{matrix} u \in L_{l o c}^{p} (τ, + \infty; L^{p} (Ω)) \cap L_{l o c}^{2} (τ, + \infty; V) \end{matrix}$ $\begin{array}{} \displaystyle u\in L_{loc}^{p}(\tau,+\infty;L^{p}(\Omega))\cap L_{loc} ^{2}(\tau,+\infty;V) \end{array}$ is called a weak solution to problem (1) on (τ, +∞) if for all T > τ, v ∈ V ∩ L^p(Ω) one has

$\begin{matrix} \frac{d}{d t} (u (t), v) + ((u (t), v)) + {(f (t, u (t)), v)}_{q, p} = (h (t), v) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left( u\left( t\right) ,v\right) +\left( \left( u\left( t\right) ,v\right) \right) +\left\langle f(t,u(t)),v\right\rangle _{q,p}=\left( h\left( t\right) ,v\right) . \end{array}$$(7)

in the sense of scalar distributions on (τ, T).

For a weak solution let g(t) = −f(t, u(t)) + h(t) + Δ u(t). From $\begin{matrix} u \in L_{l o c}^{p} (τ, + \infty; L^{p} (Ω)) \cap L_{l o c}^{2} (τ, + \infty; V) \end{matrix}$ $\begin{array}{} \displaystyle |f(t,u)|\leq C_{1}(1+|u|^{p-1}), \end{array}$ and condition (4) it is clear that $\begin{matrix} g \in L_{l o c}^{q} (τ, + \infty; L^{q} (Ω)) + L_{l o c}^{2} (τ, + \infty; V^{'}) . \end{matrix}$ $\begin{array}{} \displaystyle g\in L_{loc}^{q}(\tau,+\infty;L^{q}(\Omega))+L_{loc}^{2}(\tau,+\infty;V^{\prime}). \end{array}$ Thus, we can rewrite (7) as

$\begin{matrix} \frac{d}{d t} (u (t), v) = {(g (t), v)}_{V^{'} + L^{q}}, for all v \in V \cap L^{p} (Ω), \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left( u\left( t\right) ,v\right) =\left\langle g(t),v\right\rangle _{V^{\prime}+L^{q}}\text{, for all }v\in V\cap L^{p}(\Omega)\text{,} \end{array}$$

where 〈⋅, ⋅〉_V^′+L^q denotes pairing between V^′ + L^q(Ω) and V ∩ L^p(Ω).

It is well-known [10, p.284] that for any u_τ ∈ H there exists at least one weak solution u(⋅). If, moreover, f(t, ⋅) ∈ C¹(ℝ) and $\begin{matrix} \frac{\partial f}{\partial u} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{\partial f}{\partial u} \end{array}$ (t, u) ≥ −C for any (t, u), then the solution is unique.

In order to consider an equivalent equality to (7) we recall the following well-known result.

Lemma 1

[33, p.250] LetXbe Banach space with dualX^′, u, g ∈ L¹(a, b; X). Then the following statements are equivalent:

$\begin{matrix} u (t) = ξ + \int_{a}^{t} g (s) d s, \end{matrix}$ $\begin{array}{} u(t)=\xi+\int\limits_{a}^{t}g(s)ds, \end{array}$ξ ∈ Xfor a.a. t ∈ (a, b);

$\begin{matrix} \int_{a}^{b} u (t) φ^{'} (t) d t = - \int_{a}^{b} g (t) φ (t) d t, \forall φ \in C_{0}^{\infty} (a, b); \end{matrix}$ $\begin{array}{} \int\limits_{a}^{b}u(t)\varphi^{\prime}(t)dt=-\int\limits_{a} ^{b}g(t)\varphi(t)dt, \forall\, \varphi\in C_{0}^{\infty}(a,b); \end{array}$

$\begin{matrix} \frac{d}{d t} 〈 u, η 〉_{X^{'}, X} = 〈 g, η 〉_{X^{'}, X}, \forall η \in X^{'}, \end{matrix}$ $\begin{array}{} \displaystyle \frac{d}{dt}\langle u,\eta\rangle_{X^{\prime},X}=\langle g,\eta \rangle_{X^{\prime},X}, \forall\ \eta\in X^{\prime}, \end{array}$in the sense of scalar distributions on (a, b), where 〈⋅, ⋅〉_X^′,Xdenotes pairing betweenXandX^′.

If properties 1)-3) hold, then, moreover, u(⋅) is a.e. equivalent to a continuous function from [a, b] inX.

Applying this lemma with X = V^′ + L^q(Ω) we obtain that (7) is equivalent to the equality $\begin{matrix} \frac{d u}{d t} = g \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{du}{dt}=g \end{array}$ in the sense of X-valued distributions on every interval [τ, T]. Hence,

$\begin{matrix} \frac{d u}{d t} \in L_{l o c}^{q} (τ, + \infty; L^{q} (Ω)) + L_{l o c}^{2} (τ, + \infty; V^{'}) \end{matrix}$ $$\begin{array}{} \displaystyle \frac{du}{dt}\in L_{loc}^{q}(\tau,+\infty;L^{q}(\Omega))+L_{loc}^{2}% (\tau,+\infty;V^{\prime}) \end{array}$$(8)

and (7) is in fact equivalent to the equality

$\begin{matrix} \int_{τ}^{T} {(\frac{d u}{d t}, ξ (t))}_{V^{'} + L^{q}} d t + \int_{τ}^{T} ((u (t), ξ (t))) d t + \int_{τ}^{T} {(f (t, u (t)), ξ (t))}_{q, p} d t \\ = \int_{τ}^{T} (h (t), ξ (t)) d t, \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau}^{T}\left\langle \frac{du}{dt},\xi(t)\right\rangle _{V^{\prime }+L^{q}}dt+\int_{\tau}^{T}\left( \left( u\left( t\right) ,\xi(t)\right) \right) dt+\int_{\tau}^{T}\left\langle f(t,u(t)),\xi(t)\right\rangle _{q,p}dt\\ \displaystyle =\int_{\tau}^{T}\left( h(t),\xi(t)\right) dt, \end{array}$$(9)

for any ξ ∈ L^p(τ, T; L^p(Ω)) ∩ L²(τ, T; V).

Property (8) implies that u ∈ C([τ, +∞), H) and that the function t ↦ ∥u(t)∥² is absolutely continuous on [τ, T] and $\begin{matrix} \frac{d}{d t} ∥ u (t) ∥^{2} = 2 {(\frac{d u}{d t}, u)}_{V^{'} + L^{q}} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{d}{dt}\Vert u(t)\Vert^{2}=2\left\langle \dfrac{du}{dt},u\right\rangle _{V^{\prime}+L^{q}} \end{array}$ for a.a. t ∈ (τ, T) [10, p.285]. Hence, the initial condition u(τ) = u_τ makes sense.

Also, since the space $\begin{matrix} C_{0}^{\infty} \end{matrix}$ $\begin{array}{} \displaystyle C_{0}^{\infty} \end{array}$ ((τ, T) × Ω) is dense in L^p(τ, T; L^p(Ω)) ∩ L²(τ, T; V), equality (9) is equivalent to

$\begin{matrix} - \int_{τ}^{T} \int_{Ω} u (t, x) \frac{\partial ϕ}{\partial t} d x d t - \int_{τ}^{T} \int_{Ω} u (t, x) \frac{\partial^{2} ϕ}{\partial x^{2}} d x d t + \int_{τ}^{T} \int_{Ω} f (u (t, x)) ϕ (t, x) d x d t \\ = \int_{τ}^{T} \int_{Ω} h (t, x) ϕ (t, x) d x d t, \end{matrix}$ $$\begin{array}{} \displaystyle -\int_{\tau}^{T}\int_{\Omega}u\left( t,x\right) \frac{\partial\phi }{\partial t}dxdt-\int_{\tau}^{T}\int_{\Omega}u\left( t,x\right) \frac{\partial^{2}\phi}{\partial x^{2}}dxdt+\int_{\tau}^{T}\int_{\Omega }f\left( u\left( t,x\right) \right) \phi\left( t,x\right) dxdt\\ \displaystyle =\int_{\tau}^{T}\int_{\Omega}h\left( t,x\right) \phi\left( t,x\right) dxdt, \end{array}$$

for any ϕ ∈ $\begin{matrix} C_{0}^{\infty} \end{matrix}$ $\begin{array}{} \displaystyle C_{0}^{\infty} \end{array}$ ((τ, T) × Ω), that is, u(⋅) is a weak solution if and only if it is a solution in the sense of distributions.

Let us consider now another concept of solution. Namely, we will define a mild solution to (1).

It is known that the operator A = Δ:D(A) = H²(Ω) ∩ $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H_{0}^{1} \end{array}$ (Ω) → H is the generator of a strongly continuous semigroup of contractions T : ℝ⁺ × H → H. Moreover, it follows from well-known results [27] that for any x ∈ D(A) the function u(t) = T(t) x is the unique classical solution to the problem

$\begin{matrix} (\begin{matrix} \frac{d u}{d t} = A u (t), t > 0, \\ u (0) = x . \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{c} \dfrac{du}{dt}=Au(t),\ t \gt 0,\\ u\left( 0\right) =x. \end{array} \right. \end{array}$$(10)

Further, we shall study the inhomogeneous problem

$\begin{matrix} (\begin{matrix} \frac{d u}{d t} = A u (t) + g (t), t > τ, \\ u (τ) = x, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{c} \dfrac{du}{dt}=Au(t)+g\left( t\right) ,\, t \gt \tau,\\ u\left( \tau\right) =x, \end{array} \right. \end{array}$$(11)

where g:ℝ → H.

Definition 2

Let x ∈ H and g ∈ L¹(τ, T; H). Then the function u ∈ C([τ, T], H) is called a mild solution to problem (11) on [τ, T] if

$\begin{matrix} u (t) = T (t - τ) x + \int_{τ}^{t} T (t - s) g (s) d s, τ \leq t \leq T . \end{matrix}$ $$\begin{array}{} \displaystyle u\left( t\right) =T\left( t-\tau\right) x+\int_{\tau}^{t}T\left( t-s\right) g(s)ds\text{, }\tau\leq t\leq T. \end{array}$$(12)

If g ∈ $\begin{matrix} L_{l o c}^{1} \end{matrix}$ $\begin{array}{} \displaystyle L_{loc}^{1} \end{array}$ ([τ, +∞)), then u ∈ C([τ, +∞), H) is called a mild solution if (12) is satisfied on every interval [τ, T].

It is called a classical solution on [τ, T] if u(⋅) is continuously differentiable on (τ, T), u(t) ∈ D(A) for any t ∈ (τ, T), u(τ) = x and the equality in (11) is satisfied for every t ∈ (τ, T).

It follows readily from this definition that problem (11) possesses a unique mild solution for every x ∈ H. Also, if g is continuously differentiable on [τ, T] and x ∈ D(A), then the mild solution is in fact the unique classical solution [27, p.107].

Coming back to our problem (1), let us introduce the concept of mild solution for it.

Definition 3

The function $\begin{matrix} u \in L_{l o c}^{p} (τ, + \infty; L^{p} (Ω)) \cap L_{l o c}^{2} (τ, + \infty; V) \cap C ([τ, + \infty), H) \end{matrix}$ $\begin{array}{} \displaystyle u\in L_{loc}^{p}(\tau,+\infty;L^{p}(\Omega))\cap L_{loc} ^{2}(\tau,+\infty;V)\cap C([\tau,+\infty),H) \end{array}$ is called a mild solution to problem (1) on (τ, +∞) if for all T > τ the function g(⋅) = h(⋅) − f(⋅, u(⋅)) belongs to L¹(τ, T; H) and the equality (12) holds true.

Our aim now is to show that under an additional assumption the concepts of weak and mild solutions coincide.

Lemma 2

Assume thatu(⋅) is a weak solution to problem (1) with initial datumu_τ ∈ H which satisfies

$\begin{matrix} f (\cdot, u (\cdot)) \in L^{2} (τ, T; H) . \end{matrix}$ $$\begin{array}{} \displaystyle f\left( \cdot,u(\cdot)\right) \in L^{2}(\tau,T;H). \end{array}$$(13)

Then it is a mild solution as well.

Proof

Let us define the function g(t) = − f(t, u(t)) + h(t), which belongs to L²(τ, T; H) for every τ < T due to (13). We need to prove that the equality (12) holds. For an arbitrary fixed interval [τ, T] we take sequences $\begin{matrix} u_{0}^{n} \in H^{2} (Ω) \cap H_{0}^{1} (Ω) \end{matrix}$ $\begin{array}{} \displaystyle u_{0}^{n}\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega) \end{array}$ , gⁿ(⋅) ∈ C¹([τ, T], H), T < T, such that $\begin{matrix} u_{0}^{n} \to u_{0} \end{matrix}$ $\begin{array}{} \displaystyle u_{0} ^{n}\rightarrow u_{0} \end{array}$ in H, gⁿ → g in L²(τ, T; H). Let uⁿ be the unique classical solution to the problem

$\begin{matrix} (\begin{matrix} \frac{d u^{n}}{d t} = A u^{n} (t) + g^{n} (t), t \in (τ, \bar{T}), \\ u^{n} (τ) = u_{τ}^{n} . \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{c} \dfrac{du^{n}}{dt}=Au^{n}(t)+g^{n}\left( t\right) ,\ t\in(\tau,\overline {T}),\\ u^{n}\left( \tau\right) =u_{\tau}^{n}. \end{array} \right. \end{array}$$

Since u ∈ C¹([τ + ε, T], H) for any 0 < ε < T − τ, the difference uⁿ − u satisfies

$\begin{matrix} \frac{1}{2} \frac{d}{d t} {(u^{n} - u)}^{2} + {(u^{n} (t) - u (t))}_{V}^{2} \leq (g^{n} (t) - g (t)) (u^{n} (t) - u (t)) for a.a. t \in (τ + ε, T) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2}\frac{d}{dt}\left\Vert u^{n}-u\right\Vert ^{2}+\left\Vert u^{n}(t)-u(t)\right\Vert _{V}^{2}\leq\left\Vert g^{n}(t)-g(t)\right\Vert \left\Vert u^{n}(t)-u(t)\right\Vert \text{ for a.a. }t\in\left( \tau+\varepsilon,T\right) . \end{array}$$

Hence, by using Young’s inequality and Gronwall’s lemma it is standard to show that uⁿ → u in C([τ + ε, T], H).

Further, from the equality

$\begin{matrix} u^{n} (t) = T (t - τ - ε) u^{n} (τ + ε) + \int_{τ + ε}^{t} T (t - s) g^{n} (s) d s, τ + ε \leq t \leq T, \end{matrix}$ $$\begin{array}{} \displaystyle u^{n}\left( t\right) =T\left( t-\tau-\varepsilon\right) u^{n} (\tau+\varepsilon)+\int_{\tau+\varepsilon}^{t}T\left( t-s\right) g^{n}(s)ds\text{, }\tau+\varepsilon\leq t\leq T, \end{array}$$

and taking into account that

$\begin{matrix} T (t - τ - ε) u^{n} (τ + ε) \to T (t - τ - ε) u (τ + ε), \\ \int_{τ + ε}^{t} T (t - s) g^{n} (s) d s \to \int_{τ + ε}^{t} T (t - s) g (s) d s, \end{matrix}$ $$\begin{array}{} \displaystyle T\left( t-\tau-\varepsilon\right) u^{n}(\tau+\varepsilon)\rightarrow T\left( t-\tau-\varepsilon\right) u(\tau+\varepsilon),\\ \displaystyle\int_{\tau+\varepsilon}^{t}T\left( t-s\right) g^{n}(s)ds\rightarrow \int_{\tau+\varepsilon}^{t}T\left( t-s\right) g(s)ds, \end{array}$$

where the last convergence follows from Lebesgue’s theorem, we have

$\begin{matrix} u (t) = T (t - τ - ε) u (τ + ε) + \int_{τ + ε}^{t} T (t - s) g (s) d s, τ + ε \leq t \leq T . \end{matrix}$ $$\begin{array}{} \displaystyle u\left( t\right) =T\left( t-\tau-\varepsilon\right) u(\tau+\varepsilon )+\int_{\tau+\varepsilon}^{t}T\left( t-s\right) g(s)ds\text{, } \tau+\varepsilon\leq t\leq T. \end{array}$$

Passing to the limit as ε → 0 and using that T(t − τ − ε) u(τ + ε) → T(t − τ) u(τ) we finally obtain that

$\begin{matrix} u (t) = T (t - τ) u (τ) + \int_{τ}^{t} T (t - s) g (s) d s, τ \leq t \leq T, \end{matrix}$ $$\begin{array}{} \displaystyle u\left( t\right) =T\left( t-\tau\right) u(\tau)+\int_{\tau}^{t}T\left( t-s\right) g(s)ds\text{, }\tau\leq t\leq T, \end{array}$$

so we conclude that u(⋅) is a mild solution.

We prove now the converse statement.

Lemma 3

Assume thatu(⋅) is a mild solution to problem (1) with initial datau_τ ∈ Hwhich satisfies (13). Then it is a weak solution.

Proof

Since the function g(⋅) = h(⋅) − f(⋅, u(⋅)) belongs to L²(τ, T; H), using standard results [34, p.68] we obtain that the problem

$\begin{matrix} (\begin{matrix} \frac{\partial v}{\partial t} - Δ v = g (t, x), x \in Ω, t > τ, \\ v |_{\partial Ω} = 0, \\ v (τ, x) = u_{τ} (x), x \in Ω, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{l} \dfrac{\partial v}{\partial t}-\Delta v=g(t,x),\quad x\in\Omega,\, t \gt \tau,\\ v|_{\partial\Omega}=0,\\ v(\tau,x)=u_{\tau}(x)\text{, }x\in\Omega, \end{array} \right. \end{array}$$

possesses a unique weak solution v(⋅). However, arguing in the same way as in Lemma 2 we deduce that v(⋅) is a mild solution to problem (11) with x = u_τ. Therefore, by uniqueness of the mild solution we have u(⋅) = v(⋅), so our statement follows.

Regularity of the strong global attractor

Let us consider problem (1) in the autonomous case, that is,

$\begin{matrix} (\begin{matrix} \frac{\partial u}{\partial t} - Δ u + f (u) = h (x), x \in Ω, t > τ, \\ u |_{\partial Ω} = 0, \\ u (0, x) = u_{0} (x), x \in Ω, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array} [c]{l} \dfrac{\partial u}{\partial t}-\Delta u+f(u)=h(x),\quad x\in\Omega,\, t \gt \tau,\\ u|_{\partial\Omega}=0,\\ u(0,x)=u_{0}(x)\text{, }x\in\Omega, \end{array} \right. \end{array}$$(14)

and we assume that conditions (2)-(5) are satisfied. In particular, this means that h ∈ H.

In [17] the existence of a global attractor for the multivalued semiflow generated by the strong solutions to (14) was proved assuming a critical growth condition on the nonlinear function f. In this section, our aim is to prove that this attractor is bounded in suitable L^r-spaces.

Definition 4

A weak solution u(⋅) to problem (1) is called a strong solution if, additionally, it satisfies

$\begin{matrix} u \in L_{l o c}^{\infty} (0, + \infty; V \cap L^{p} (Ω)), \end{matrix}$ $$\begin{array}{} \displaystyle u\in L_{loc}^{\infty}\left( 0,+\infty;V\cap L^{p}(\Omega)\right) , \end{array}$$(15)

$\begin{matrix} \frac{d u}{d t} \in L_{l o c}^{2} (0, + \infty; H) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{du}{dt}\in L_{loc}^{2}\left( 0,+\infty;H\right) . \end{array}$$(16)

As u ∈ L^∞(0, T; V ∩ L^p(Ω)) ∩ C([0, T]; H), we deduce that $\begin{matrix} u \in C ([0, T]; H_{0 w}^{1} (Ω) \cap L_{w}^{p} (Ω)) \end{matrix}$ $\begin{array}{} \displaystyle u\in C([0,T];H_{0w}^{1}(\Omega)\cap L_{w}^{p}(\Omega)) \end{array}$ for any T > 0, where $\begin{matrix} H_{0 w}^{1} (Ω), L_{w}^{p} (Ω) \end{matrix}$ $\begin{array}{} \displaystyle H_{0w}^{1}(\Omega),\, L_{w}^{p}(\Omega) \end{array}$ are respectively the spaces $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H_{0}^{1} \end{array}$ (Ω), L^p(Ω) with the weak topology [33, Lemma 1.4, p.263].

Throughout this section we will assume that

$\begin{matrix} p \leq \frac{2 N - 2}{N - 2} \end{matrix}$ $$\begin{array}{} \displaystyle p\leq\frac{2N-2}{N-2} \end{array}$$(17)

if N ≥ 3, that is, f satisfies a critical growth condition. No assumption is imposed if N ≤2.

From (4), (17) and the imbeddings H¹(Ω) ⊂ $\begin{matrix} L^{\frac{2 N}{N - 2}} \end{matrix}$ $\begin{array}{} \displaystyle L^{\frac{2N}{N-2}} \end{array}$ (Ω), if N ≥ 3, H¹(Ω) ⊂ L^q(Ω), for any q ≥ 1 if N ≤2, we get

$\begin{matrix} \int_{τ}^{T} \int_{Ω} {(f (u))}^{2} d x d t \leq K_{1} \int_{τ}^{T} (1 + {(u (t))}_{L^{2 p - 2} (Ω)}^{2 p - 2}) d t \leq K_{2} \int_{τ}^{T} (1 + {(u (t))}_{V}^{2 p - 2}) d t . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau}^{T}\int_{\Omega}\left\vert f\left( u\right) \right\vert ^{2}dxdt\leq K_{1}\int_{\tau}^{T}\left( 1+\left\Vert u\left( t\right) \right\Vert _{L^{2p-2}(\Omega)}^{2p-2}\right) dt\leq K_{2}\int_{\tau} ^{T}\left( 1+\left\Vert u\left( t\right) \right\Vert _{V}^{2p-2}\right) dt. \end{array}$$(18)

Then the $\begin{matrix} Δ u = \frac{d u}{d t} + f (u) - h \end{matrix}$ $\begin{array}{} \displaystyle \Delta u=\dfrac{du}{dt}+f\left( u\right) -h \end{array}$ and (15)-(16) imply that

$\begin{matrix} u \in L^{2} (τ, T; D (A)) \end{matrix}$ $$\begin{array}{} \displaystyle u\in L^{2}\left(\tau,T;D\left(A\right)\right) \end{array}$$(19)

for any strong solution u(⋅).

Since $\begin{matrix} p \leq \frac{2 N - 2}{N - 2} \leq \frac{2 N}{N - 2}, \end{matrix}$ $\begin{array}{} \displaystyle p\leq\frac{2N-2}{N-2}\leq\frac{2N}{N-2}, \end{array}$ it is also obvious that V ⊂ L^p(Ω).

By standard results [28, Corollary 7.3] we obtain then that u ∈ C([0, +∞), V) and

$\begin{matrix} \frac{d}{d t} {(u)}_{V}^{2} = 2 (- Δ u, \frac{d u}{d t}) for a.a. t > 0. \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left\Vert u\right\Vert _{V}^{2}=2\left( -\Delta u,\frac{du} {dt}\right) \text{ for a.a. }t \gt 0. \end{array}$$(20)

It is known [17] that for any u₀ ∈ V there exists at least one strong solution u(⋅) such that u(0) = u₀. Moreover, every strong solution satisfies the energy inequality

$\begin{matrix} E (u (t)) + 2 \int_{s}^{t} {(u_{r})}^{2} d r = E (u (s)), for all t \geq s \geq 0, \end{matrix}$ $$\begin{array}{} \displaystyle E\left( u\left( t\right) \right) +2\int_{s}^{t}\left\Vert u_{r}\right\Vert ^{2}dr=E\left( u\left( s\right) \right) \text{, for all }t\geq s\geq0, \end{array}$$(21)

where E(u(t)) = $\begin{matrix} {(u (t))}_{V}^{2} \end{matrix}$ $\begin{array}{} \displaystyle \left\Vert u\left( t\right) \right\Vert _{V}^{2} \end{array}$ + 2(F(u(t)), 1) − 2(h, u(t)).

Let

$\begin{matrix} K_{s}^{+} = {u (\cdot) : u is a strong solution of (14)} . \end{matrix}$ $$\begin{array}{} \displaystyle K_{s}^{+}=\{u\left(\cdot\right) :u\text{ is a strong solution of (14)}\}. \end{array}$$

We define now the multivalued map G : ℝ⁺ × V → P(V), where P(V) is the set of all non-empty subsets of V, by

$\begin{matrix} G (t, u_{0}) = {u (t) : u \in K_{s}^{+} and u (0) = u_{0}} . \end{matrix}$ $$\begin{array}{} \displaystyle G(t,u_{0})=\{u\left( t\right) :u\in K_{s}^{+}\text{ and }u\left( 0\right) =u_{0}\}. \end{array}$$

We recall now some results proved in [17] about the existence and structure of a global attractor for G. It is worth pointing out that, although in that paper such theorems were proved in the particular three-dimensional case (i.e. N = 3), the proofs in the general N-dimensional setting are identical.

First, we note that G is a strict multivalued semiflow, that is, G(0, x) = x and G(t + s, x) = G(t, G(s, x)) for any t, s ∈ ℝ⁺, x ∈ V.

Moreover, G possesses a global compact invariant attractor 𝓐, that is, 𝓐 is compact in V, it is invariant (i.e. 𝓐 = G(t, 𝓐) for any t ≥ 0) and attracts every bounded set in V, that is,

$\begin{matrix} d i s t (G (t, B), A) \to 0 as t \to + \infty, \end{matrix}$ $$\begin{array}{} \displaystyle dist(G(t,B),\mathscr{A)}\rightarrow0\text{ as }t\rightarrow+\infty\text{, } \end{array}$$

for any B set bounded in V, where dist refers to the Hausdorff semidistance between sets given by dist(A, B) = sup_x∈A inf_y∈B ∥x − y∥_V.

The map γ : ℝ → V is called a complete trajectory of $\begin{matrix} K_{s}^{+} \end{matrix}$ $\begin{array}{} \displaystyle K_{s}^{+} \end{array}$ if γ(⋅ + h)|_[0,+∞) ∈ $\begin{matrix} K_{s}^{+} \end{matrix}$ $\begin{array}{} \displaystyle K_{s}^{+} \end{array}$ for any h ∈ ℝ, and this is equivalent to γ being continuous and satisfying

$\begin{matrix} γ (t + s) \in G_{s} (t, γ (s)) for all s \in R and t \geq 0. \end{matrix}$ $$\begin{array}{} \displaystyle \gamma\left( t+s\right) \in G_{s}\left( t,\gamma\left( s\right) \right) \text{ for all }s\in\mathbb{R}\text{ and }t\geq0. \end{array}$$(22)

The set of all complete trajectories of $\begin{matrix} K_{s}^{+} \end{matrix}$ $\begin{array}{} \displaystyle K_{s}^{+} \end{array}$ will be denoted by 𝔽_s. Let 𝕂_s be the set of all complete trajectories which are bounded in V.

The attractor 𝓐 is characterised by the union of all points lying in a bounded complete trajectory, that is,

$\begin{matrix} A = {γ (0) : γ (\cdot) \in K_{s}} = \cup_{t \in R} {γ (t) : γ (\cdot) \in K_{s}} . \end{matrix}$ $$\begin{array}{} \displaystyle \mathscr{A}=\{\gamma\left( 0\right) :\gamma\left( \cdot\right) \in\mathbb{K}_{s}\}=\cup_{t\in\mathbb{R}}\{\gamma\left( t\right) :\gamma\left( \cdot\right) \in\mathbb{K}_{s}\}. \end{array}$$(23)

A point z ∈ X is a fixed point of $\begin{matrix} K_{s}^{+} \end{matrix}$ $\begin{array}{} \displaystyle K_{s}^{+} \end{array}$ if φ(t) ≡ z ∈ $\begin{matrix} K_{s}^{+} \end{matrix}$ $\begin{array}{} \displaystyle K_{s}^{+} \end{array}$ , whereas it is called a fixed point of G if z ∈ G(t, z) for all t ≥ 0. In our case these two concepts are equivalent, so we will simply call them fixed points. Moreover, z is a fixed point if and only if z ∈ V ∩ H²(Ω) and

$\begin{matrix} - Δ z + f (z) = h in H . \end{matrix}$ $$\begin{array}{} \displaystyle -\Delta z+f(z)=h\text{ in }H. \end{array}$$(24)

The set of all fixed points will be denoted by ℜ.

Finally, in [17] it is proved that the strong global attractor coincides with the unstable manifold of the set of stationary points, and also with the stable one when we consider only bounded complete solutions. This means that

$\begin{matrix} Θ_{s} = M_{s}^{+} (R) = M_{s}^{-} (R), \end{matrix}$ $$\begin{array}{} \displaystyle \Theta_{s}=M_{s}^{+}(\mathfrak{R})=M_{s}^{-}(\mathfrak{R}), \end{array}$$(25)

where

$\begin{matrix} M_{s}^{-} (R) = (z : \exists γ (\cdot) \in K_{s}, γ (0) = z, d i s t (γ (t), R) \to 0, t \to + \infty), \\ M_{s}^{+} (R) = (z : \exists γ (\cdot) \in F_{s}, γ (0) = z, d i s t (γ (t), R) \to 0, t \to - \infty) . \end{matrix}$ $$\begin{array}{} \displaystyle M_{s}^{-}(\mathfrak{R})=\left\{ z\,:\,\exists\gamma(\cdot)\in\mathbb{K} _{s},\,\,\gamma(0)=z,\,\,\,\, dist(\gamma(t),\mathfrak{R})\rightarrow 0,\,\, t\rightarrow+\infty\right\} ,\\ M_{s}^{+}(\mathfrak{R})=\left\{ z\,:\,\exists\gamma(\cdot)\in\mathbb{F} _{s},\,\, \gamma(0)=z,\,\,\,\,dist(\gamma(t),\mathfrak{R})\rightarrow 0,\,\, t\rightarrow-\infty\right\} . \end{array}$$

Concerning boundedness of the attractor in stronger L^r-spaces than $\begin{matrix} L^{\frac{2 N}{N - 2}} \end{matrix}$ $\begin{array}{} \displaystyle L^{\frac{2N}{N-2}} \end{array}$ (Ω) (which follows from the embedding V ⊂ $\begin{matrix} L^{\frac{2 N}{N - 2}} \end{matrix}$ $\begin{array}{} \displaystyle L^{\frac{2N}{N-2}} \end{array}$ (Ω)) in [17] and [18] an estimate in the space L^∞(Ω) is shown to be true if h ∈ L^∞(Ω).

Now, using the equivalence between weak and mild solutions from the first section we are able to obtain suitable estimates in L^r-spaces under much weaker assumptions on the function h.

Theorem 4

The global attractor 𝓐 is bounded inL^r(Ω), wherer ∈ [1, +∞] ifN ≤3, r ∈ [1, +∞) ifN = 4 and 1 ≤ r < $\begin{matrix} \frac{2 N}{N - 4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{2N}{N-4} \end{array}$ifN ≥ 5.

IfN ≥ 4 andh ∈ L^q(Ω), for someq > $\begin{matrix} \frac{N}{2} \end{matrix}$ $\begin{array}{} \displaystyle \frac{N}{2} \end{array}$ , then 𝓐 is bounded inL^∞(Ω).

Proof

The semigroup T(t) generated by the solutions of problem (10) satisfies the following well-known estimate [32, p.2988]:

$\begin{matrix} {(T (t) x)}_{L^{r} (Ω)} \leq M \frac{e^{δ t}}{t^{\frac{N}{2} (\frac{1}{q} - \frac{1}{r})}} {(x)}_{L^{q} (Ω)}, \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert T(t)x\right\Vert _{L^{r}(\Omega)}\leq M\frac{e^{\delta t}} {t^{\frac{N}{2}\left( \frac{1}{q}-\frac{1}{r}\right) }}\left\Vert x\right\Vert _{L^{q}(\Omega)}, \end{array}$$(26)

for every 1 ≤ q ≤ r ≤∞ and certain M > 0, δ ∈ ℝ.

Take an arbitrary y ∈ 𝓐. In view of (23), there exists a complete trajectory ψ such that ψ(0) = y and ψ(t) ∈ 𝓐 for all t ∈ ℝ. Inequality (18), combined with (15) and Lemma 2, implies that u(⋅) = ψ(⋅ + h) is a mild solution to problem (14) for any h ∈ ℝ. We choose h = −1 and then by the variation of constants formula we have

$\begin{matrix} y = u (1) = T (1) u (0) + \int_{0}^{1} T (1 - s) (h - f (u (s)) d s . \end{matrix}$ $$\begin{array}{} \displaystyle y=u(1)=T\left( 1\right) u\left( 0\right) +\int_{0}^{1}T(1-s)\left( h-f(u(s)\right) ds. \end{array}$$

Hence, applying (26) with q = 2 it follows that

$\begin{matrix} {(y)}_{L^{r} (Ω)} \leq M e^{δ} (u (0)) + \int_{0}^{1} M \frac{e^{δ s}}{{(1 - s)}^{\frac{N}{2} (\frac{1}{2} - \frac{1}{r})}} ((h) + (f (u (s))) d s \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert y\right\Vert _{L^{r}(\Omega)}\leq Me^{\delta}\left\Vert u\left( 0\right) \right\Vert +\int_{0}^{1}M\frac{e^{\delta s}}{\left( 1-s\right) ^{\frac{N}{2}\left( \frac{1}{2}-\frac{1}{r}\right) }}\left( \left\Vert h\right\Vert +\left\Vert f(u(s)\right\Vert \right) ds \end{array}$$

for any r ≥ 2. Take an arbitrary value r satisfying r ∈ [2, +∞] if N ≤3, r ∈ [2, +∞) if N = 4 and 2 ≤ r < $\begin{matrix} \frac{2 N}{N - 4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{2N}{N-4} \end{array}$ if N ≥ 5. The inequality ∥f(u(s)∥² ≤ R₁(1 + $\begin{matrix} {(u (t))}_{V}^{2 p - 2} \end{matrix}$ $\begin{array}{} \displaystyle \left\Vert u\left( t\right) \right\Vert _{V}^{2p-2} \end{array}$ ), together with the boundedness of the attractor in the space V, gives the existence of a constant R₂ such that

$\begin{matrix} {(y)}_{L^{r} (Ω)} \leq R_{2} (1 + \int_{0}^{1} \frac{1}{{(1 - s)}^{\frac{N}{2} (\frac{1}{2} - \frac{1}{r})}} d s) = R_{2} (1 + \frac{1}{\frac{N}{2} (\frac{1}{r} - \frac{1}{2}) + 1}), \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert y\right\Vert _{L^{r}(\Omega)}\leq R_{2}\left( 1+\int_{0}^{1} \frac{1}{\left( 1-s\right) ^{\frac{N}{2}\left( \frac{1}{2}-\frac{1} {r}\right) }}ds\right) =R_{2}\left( 1+\frac{1}{\frac{N}{2}\left( \frac {1}{r}-\frac{1}{2}\right) +1}\right) , \end{array}$$

where we have used that due to the conditions imposed on the parameter r, it follows that $\begin{matrix} \frac{N}{2} (\frac{1}{r} - \frac{1}{2}) + 1 > 0 \end{matrix}$ $\begin{array}{} \displaystyle \frac{N}{2}\left( \frac{1}{r}-\frac{1}{2}\right) +1 \gt 0 \end{array}$.

Thus the first statement of the theorem is proved.

In order to prove the second one, we first will prove that if the global attractor is bounded in L^s(Ω), where s satisfies

$\begin{matrix} s \frac{N - 2}{N} > \frac{N}{2}, \end{matrix}$ $$\begin{array}{} \displaystyle s\frac{N-2}{N} \gt \frac{N}{2}, \end{array}$$(27)

then it is bounded in fact in L^∞(Ω). By (4) and (17) we get

$\begin{matrix} \int_{Ω} {(f (u (s, x))}^{s \frac{N - 2}{N}} d x \leq R_{3} (1 + \int_{Ω} {(u (s, x))}^{s \frac{N - 2}{N} (p - 1)} d x) \\ \leq R_{3} (1 + \int_{Ω} {(u (s, x))}^{s} d x) \leq R_{4} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\Omega}\left\vert f(u(s,x)\right\vert ^{s\frac{N-2}{N}}dx \leq R_{3}(1+\int_{\Omega}\left\vert u(s,x)\right\vert ^{s\frac{N-2}{N}\left( p-1\right) }dx)\\\displaystyle \qquad\qquad\qquad\qquad~~ \leq R_{3}(1+\int_{\Omega}\left\vert u(s,x)\right\vert ^{s}dx)\leq R_{4}. \end{array}$$(28)

Therefore, applying again formula (26) with r = ∞ and $\begin{matrix} q = min {\bar{q}, s \frac{N - 2}{N}} \end{matrix}$ $\begin{array}{} \displaystyle q=\min\{\overline{q},s\frac{N-2}{N}\} \end{array}$ and arguing as before for any y ∈ 𝓐 we have

$\begin{matrix} {(y)}_{L^{\infty} (Ω)} \leq M e^{δ} {(u (0))}_{L^{q} (Ω)} + \int_{0}^{1} M \frac{e^{δ s}}{{(1 - s)}^{\frac{N}{2} \frac{1}{q}}} ({(h)}_{L^{q} (Ω)} + {(f (u (s))}_{L^{q} (Ω)}) d s \\ \leq R_{5} (1 + \int_{0}^{1} \frac{1}{{(1 - s)}^{\frac{N}{2} \frac{1}{q}}} d s) = R_{5} (1 + \frac{1}{- \frac{N}{2 q} + 1}), \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert y\right\Vert _{L^{\infty}(\Omega)} \leq Me^{\delta}\left\Vert u\left( 0\right) \right\Vert _{L^{q}(\Omega)}+\int_{0}^{1}M\frac{e^{\delta s}}{\left( 1-s\right) ^{\frac{N}{2}\frac{1}{q}}}\left( \left\Vert h\right\Vert _{L^{q}(\Omega)}+\left\Vert f(u(s)\right\Vert _{L^{q}(\Omega )}\right) ds\\ \displaystyle\qquad\qquad \leq R_{5}\left( 1+\int_{0}^{1}\frac{1}{\left( 1-s\right) ^{\frac{N} {2}\frac{1}{q}}}ds\right) =R_{5}\left( 1+\frac{1}{-\frac{N}{2q}+1}\right) , \end{array}$$

where we have used that $\begin{matrix} - \frac{N}{2 q} + 1 > 0. \end{matrix}$ $\begin{array}{} \displaystyle -\frac{N}{2q}+1 \gt 0. \end{array}$ Thus, the result follows.

Observe that if N = 4, then the attractor is bounded in L^s(Ω) for an arbitrary s ∈ [2, ∞). Hence, we can pick s such that (27) holds. On the other hand, since we have proved that the attractor is bounded in L^s(Ω) for any s < $\begin{matrix} \frac{2 N}{N - 4} \end{matrix}$ $\begin{array}{} \displaystyle \frac{2N}{N-4} \end{array}$ , then (27) is also satisfied if N = 5, 6.

Further, for N ≥ 7 we will apply the above procedure iteratively so as to achieve (27).

Assume that the attractor is bounded in L^s(Ω) for some $\begin{matrix} s \geq \frac{2 N}{N - 2} such that s \frac{N - 2}{N} < \frac{N}{2} . \end{matrix}$ $\begin{array}{} \displaystyle s\geq \frac{2N}{N-2}\,\, \text{such that}\,\, s\frac{N-2}{N} \lt \frac{N}{2}. \end{array}$ Using (28) and applying again (26) with $\begin{matrix} q = s \frac{N - 2}{N} and \bar{s} < \frac{s (N - 2) N}{N^{2} - 2 s (N - 2)} \end{matrix}$ $\begin{array}{} \displaystyle q=s\frac{N-2}{N}\,\, \text{and}\,\, \overline {s} \lt \frac{s(N-2)N}{N^{2}-2s(N-2)} \end{array}$ for any y ∈ 𝓐 we obtain

$\begin{matrix} {(y)}_{L^{\bar{s}} (Ω)} \leq R_{6} (1 + \int_{0}^{1} \frac{1}{{(1 - s)}^{\frac{N}{2} (\frac{N}{r (N - 2)} - \frac{1}{\bar{s}})}} d s) = \frac{R_{6}}{\frac{N}{2} (\frac{1}{\bar{s}} - \frac{N}{s (N - 2)}) + 1} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert y\right\Vert _{L^{\overline{s}}(\Omega)}\leq R_{6}\left( 1+\int_{0}^{1}\frac{1}{\left( 1-s\right) ^{\frac{N}{2}\left( \frac {N}{r\left( N-2\right) }-\frac{1}{\overline{s}}\right) }}ds\right) =\frac{R_{6}}{\frac{N}{2}\left( \frac{1}{\overline{s}}-\frac{N}{s\left( N-2\right) }\right) +1}. \end{array}$$

Therefore, the attractor is bounded in L^s(Ω) as well. For an arbitrary ε > 0 we choose s such that the difference s − s satisfies

$\begin{matrix} \bar{s} - s \geq \frac{s (N - 2) N}{N^{2} - 2 s (N - 2)} - s - ε . \end{matrix}$ $$\begin{array}{} \displaystyle \overline{s}-s\geq\frac{s(N-2)N}{N^{2}-2s(N-2)}-s-\varepsilon. \end{array}$$

Since $\begin{matrix} s \geq \frac{2 N}{N - 2}, \end{matrix}$ $\begin{array}{} \displaystyle s\geq\frac{2N}{N-2}, \end{array}$ we get

$\begin{matrix} \bar{s} - s \geq s (\frac{(N - 2) N}{N^{2} - 4 N} - 1) - ε = s \frac{2}{N - 4} - ε \geq \frac{4 N}{(N - 2) (N - 4)} - ε . \end{matrix}$ $$\begin{array}{} \displaystyle \overline{s}-s\geq s\left( \frac{(N-2)N}{N^{2}-4N}-1\right) -\varepsilon =s\frac{2}{N-4}-\varepsilon\geq\frac{4N}{\left( N-2\right) \left( N-4\right) }-\varepsilon. \end{array}$$

There exist ε > 0 and k ∈ ℕ such that

$\begin{matrix} \frac{2 N}{N - 2} + (k - 1) (\frac{4 N}{(N - 2) (N - 4)} - ε) < \frac{N^{2}}{2 (N - 2)} < \frac{2 N}{N - 2} + k (\frac{4 N}{(N - 2) (N - 4)} - ε) \end{matrix}$ $$\begin{array}{} \displaystyle \frac{2N}{N-2}+\left( k-1\right) \left( \frac{4N}{\left( N-2\right) \left( N-4\right) }-\varepsilon\right) \lt \frac{N^{2}}{2(N-2)} \lt \frac{2N} {N-2}+k\left( \frac{4N}{\left( N-2\right) \left( N-4\right) } -\varepsilon\right) \end{array}$$

and thus proceeding iteratively we obtain in k steps that the global attractor is bounded in L^s(Ω), where s satisfies (27).

Weak attractor for strong solutions in the supercritical case

In this section, our aim is to prove the existence of a weak global attractor for the multivalued semiflow generated by strong solutions to problem (14) satisfying a suitable energy inequality without imposing the assumption (17). In this case, we do not know whether strong solutions belong to the space of continuous functions with values in V, and therefore we are still not able to prove the existence of a strong attractor. Instead, we have to consider a weaker attractor in which the attracting property is satisfied with respect to the weak topology of the space V ∩ L^p(Ω).

Lemma 5

Letu_τ ∈ V ∩ L^p(Ω). Then there exists at least one strong solutionuof (1) such thatu(τ) = u_τ. Moreover, the energy inequality

$\begin{matrix} E (u (t)) + \int_{s}^{t} {(\frac{d u}{d r})}^{2} d r \leq E (u (s)) \end{matrix}$ $$\begin{array}{} \displaystyle E(u(t))+\int_{s}^{t}\left\Vert \frac{du}{dr}\right\Vert ^{2}dr\leq E(u(s)) \end{array}$$(29)

holds for allt ≥ s, a.a. s > 0 ands = 0, where $\begin{matrix} E (u (t)) = {(u (t))}_{V}^{2} + (F (u (t)), 1) - 2 (h, u (t)) . \end{matrix}$ $\begin{array}{} \displaystyle E(u(t))=\left\Vert u\left( t\right) \right\Vert _{V}^{2}+\left( F(u(t)),1\right) -2\left( h,u\left( t\right) \right) . \end{array}$

Proof

As usual, we take the Galerkin approximations using the basis of eigenfunctions {w_j(x), j ∈ ℕ} of −Δ with Dirichlet boundary conditions. Let X_m = span{w₁, …, w_m} and let P_m be the orthogonal projector from H onto X_m. Then $\begin{matrix} u_{m} (t, x) = \sum_{j = i}^{m} a_{j, m} (t) w_{j} (x) \end{matrix}$ $\begin{array}{} u_{m}\left( t,x\right) =\sum_{j=i}^{m}a_{j,m}\left( t\right) w_{j}\left( x\right) \end{array}$ will be a solution of the system of ordinary differential equations

$\begin{matrix} \frac{d u_{m}}{d t} = P_{m} Δ u_{m} - P_{m} f (u_{m}) + P_{m} h, u_{m} (0) = P_{m} u_{0} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{du_{m}}{dt}=P_{m}\Delta u_{m}-P_{m}f\left( u_{m}\right) +P_{m} h,\, u_{m}\left( 0\right) =P_{m}u_{0}. \end{array}$$(30)

It is proved in [10, p.281] that passing to a subsequence u_m converges to a weak solution u of (1) weakly star in L^∞(0, T; H), weakly in L^p(0, T; L^p(Ω)) and weakly in L²(0, T; V) for all T > 0. Also, u(τ) = u_τ and $\begin{matrix} \frac{d u_{m}}{d t} \to \frac{d u}{d t} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{du_{m}} {dt}\rightarrow\dfrac{du}{dt} \end{array}$ weakly in L^q(0, T; H^−s(Ω)) for some s > 0.

Multiplying (30) by $\begin{matrix} \frac{d u_{m}}{d t} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{du_{m}}{dt} \end{array}$ we get

$\begin{matrix} \frac{d}{d t} (∥ u_{m} ∥_{V}^{2} + 2 (F (u_{m}), 1) - 2 (h, u_{m})) + 2 ∥ \frac{d u_{m}}{d t} ∥^{2} = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left( \Vert u_{m}\Vert_{V}^{2}+2(F(u_{m}),1)-2\left( h,u_{m}\right) \right) +2\Vert\frac{du_{m}}{dt}\Vert^{2}=0, \end{array}$$(31)

so (6) implies

$\begin{matrix} ∥ u_{m} (t) ∥_{V}^{2} + 2 \int_{0}^{t} ∥ \frac{d}{d s} u_{m} (s) ∥^{2} d s + 2 δ {(u (t))}_{L^{p} (Ω)}^{p} \\ \leq ∥ u_{m} (0) ∥_{V}^{2} + R_{1} ∥ u_{m} (0) ∥_{L^{p} (Ω)}^{p} + 2 (h) (u_{m} (t)) + 2 (h) (u_{m} (0)) + R_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad~~\Vert u_{m}(t)\Vert_{V}^{2}+2\int\limits_{0}^{t}\Vert\frac{d}{ds}u_{m} (s)\Vert^{2}ds+2\delta\left\Vert u\left( t\right) \right\Vert _{L^{p} (\Omega)}^{p} \\ \leq\Vert u_{m}\left( 0\right) \Vert_{V}^{2}+R_{1}\Vert u_{m}\left( 0\right) \Vert_{L^{p}(\Omega)}^{p}+2\left\Vert h\right\Vert \left\Vert u_{m}(t)\right\Vert +2\left\Vert h\right\Vert \left\Vert u_{m}(0)\right\Vert +R_{2}. \end{array}$$

If p > 2, δ > 0 implies that

$\begin{matrix} ∥ u_{m} (t) ∥_{V}^{2} + \int_{0}^{t} ∥ \frac{d}{d s} u_{m} (s) ∥^{2} d s + {(u (t))}_{L^{p} (Ω)}^{p} \leq R_{3} (∥ u_{m} (0) ∥_{V}^{2} + ∥ u_{m} (0) ∥_{L^{p} (Ω)}^{p}) + R_{4}, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_{m}\left( t\right) \Vert_{V}^{2}+\int\limits_{0}^{t}\Vert\frac {d}{ds}u_{m}(s)\Vert^{2}ds+\left\Vert u\left( t\right) \right\Vert _{L^{p}(\Omega)}^{p}\leq R_{3}\left( \Vert u_{m}\left( 0\right) \Vert _{V}^{2}+\Vert u_{m}\left( 0\right) \Vert_{L^{p}(\Omega)}^{p}\right) +R_{4}, \end{array}$$

where R_j > 0. When p = 2 we obtain that

$\begin{matrix} ∥ u_{m} (t) ∥_{V}^{2} + \int_{0}^{t} ∥ \frac{d}{d s} u_{m} (s) ∥^{2} d s \leq R_{3} ∥ u_{m} (0) ∥_{V}^{2} + R_{4} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_{m}\left( t\right) \Vert_{V}^{2}+\int\limits_{0}^{t}\Vert\frac {d}{ds}u_{m}(s)\Vert^{2}ds\leq R_{3}\Vert u_{m}\left( 0\right) \Vert_{V} ^{2}+R_{4}. \end{array}$$

By the choise of the special basis we have that u_m(0) → u₀ in $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H_{0}^{1} \end{array}$ (Ω) ∩ L^p(Ω).

Hence, u_m → u weakly star in L^∞(0, T; V ∩ L^p (Ω)) and $\begin{matrix} \frac{d u_{m}}{d t} \to \frac{d u}{d t} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{du_{m}}{dt}\rightarrow\dfrac{du}{dt} \end{array}$ weakly in L²(0, T; L²(Ω)). Thus from the Ascoli-Arzelà theorem {u_m} is pre-compact in C([0, T]; H) and then u_m → u in C([0, T]; H). Moreover, it follows that u_m (t) → u(t) in $\begin{matrix} H_{0 w}^{1} (Ω) \cap L_{w}^{p} (Ω)) \end{matrix}$ $\begin{array}{} \displaystyle H_{0w}^{1}(\Omega)\cap L_{w}^{p}(\Omega)) \end{array}$ for any t ∈ [0, T].

Finally, we must check the validity of the energy inequality (29). It is clear from (31) that u_m satisfy

$\begin{matrix} E (u_{m} (t)) + \int_{s}^{t} {(\frac{d u_{m}}{d r})}^{2} d r \leq E (u_{m} (s)) \end{matrix}$ $$\begin{array}{} \displaystyle E(u_{m}(t))+\int_{s}^{t}\left\Vert \frac{du_{m}}{dr}\right\Vert ^{2}dr\leq E(u_{m}(s)) \end{array}$$

for all t ≥ s ≥ 0.

Let us define the function

$\begin{matrix} L (u) = f (u) u - α | u |^{p} . \end{matrix}$ $$\begin{array}{} \displaystyle L(u)=f(u)u-\alpha|u|^{p}. \end{array}$$

Multiplying the equation in (14) by tu_m(t) and integrating over (0, T) × Ω we obtain

$\begin{matrix} \int_{0}^{T} \int_{Ω} t \frac{d u_{m}}{d t} u_{m} d x d t + \int_{0}^{T} t {(u_{m} (t))}_{V}^{2} d t \\ + \int_{0}^{T} \int_{Ω} t f (u_{m} (t, x)) u_{m} (t, x) d x d t - \int_{0}^{T} \int_{Ω} t h (x) u_{m} (t, x) d x d t = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \int_{0}^{T}\int_{\Omega}t\frac{du_{m}}{dt}u_{m}dxdt+\int_{0} ^{T}t\left\Vert u_{m}(t)\right\Vert _{V}^{2}dt\\ \displaystyle +\int_{0}^{T}\int_{\Omega}tf\left( u_{m}\left( t,x\right) \right) u_{m}\left( t,x\right) dxdt-\int_{0}^{T}\int_{\Omega}th\left( x\right) u_{m}\left( t,x\right) dxdt=0, \end{array}$$

$\begin{matrix} \frac{T}{2} {(u_{m} (T))}^{2} + \int_{0}^{T} t {(u_{m} (t))}_{V}^{2} d t \\ + \int_{0}^{T} \int_{Ω} t L (u_{m} (t, x)) d x d t + α \int_{0}^{T} t {(u_{m} (t))}_{L^{p} (Ω)}^{p} d t \\ = \frac{1}{2} \int_{0}^{T} {(u_{m} (t))}^{2} d t + \int_{0}^{T} \int_{Ω} t h (x) u_{m} (t, x) d x d t . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{T}{2}\left\Vert u_{m}\left( T\right) \right\Vert ^{2}+\int_{0} ^{T}t\left\Vert u_{m}(t)\right\Vert _{V}^{2}dt\\ \displaystyle +\int_{0}^{T}\int_{\Omega}tL(u_{m}(t,x))dxdt+\alpha\int_{0}^{T}t\left\Vert u_{m}\left( t\right) \right\Vert _{L^{p}(\Omega)}^{p}dt\\ \displaystyle =\frac{1}{2}\int_{0}^{T}\left\Vert u_{m}\left( t\right) \right\Vert ^{2}dt+\int_{0}^{T}\int_{\Omega}th\left( x\right) u_{m}\left( t,x\right) dxdt. \end{array}$$(32)

In the same way, for the limit function u we obtain

$\begin{matrix} \frac{T}{2} {(u (T))}^{2} + \int_{0}^{T} t {(u (t))}_{V}^{2} d t \\ + \int_{0}^{T} \int_{Ω} t L (u (t, x)) d x d t + α \int_{0}^{T} t {(u (t))}_{L^{p} (Ω)}^{p} d t \\ = \frac{1}{2} \int_{0}^{T} {(u (t))}^{2} d t + \int_{0}^{T} \int_{Ω} t h (x) u (t, x) d x d t . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{T}{2}\left\Vert u\left( T\right) \right\Vert ^{2}+\int_{0} ^{T}t\left\Vert u(t)\right\Vert _{V}^{2}dt\\ \displaystyle +\int_{0}^{T}\int_{\Omega}tL(u(t,x))dxdt+\alpha\int_{0}^{T}t\left\Vert u\left( t\right) \right\Vert _{L^{p}(\Omega)}^{p}dt\\ \displaystyle =\frac{1}{2}\int_{0}^{T}\left\Vert u\left( t\right) \right\Vert ^{2}dt+\int_{0}^{T}\int_{\Omega}th\left( x\right) u\left( t,x\right) dxdt. \end{array}$$(33)

From the previous convergences it is clear that

$\begin{matrix} \int_{0}^{T} {(u_{m} (t))}^{2} d t \to \int_{0}^{T} {(u (t))}^{2} d t, \\ \int_{0}^{T} \int_{Ω} t h (x) u_{m} (t, x) d x d t \to \int_{0}^{T} \int_{Ω} t h (x) u (t, x) d x d t, \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\int_{0}^{T}\left\Vert u_{m}\left( t\right) \right\Vert ^{2}dt \rightarrow\int_{0}^{T}\left\Vert u\left( t\right) \right\Vert ^{2}dt,\\ \displaystyle\int_{0}^{T}\int_{\Omega}th\left( x\right) u_{m}\left( t,x\right) dxdt \rightarrow\int_{0}^{T}\int_{\Omega}th\left( x\right) u\left( t,x\right)dxdt, \end{array}$$

which implies that the left-hand side of (32) converges to the left-hand side of (33). On the other hand, we have

$\begin{matrix} \int_{0}^{T} \int_{Ω} t L (u (t, x)) d x d t \leq lim inf \int_{0}^{T} \int_{Ω} t L (u_{m} (t, x)) d x d t, \end{matrix}$ $$\begin{array}{} \displaystyle \int_{0}^{T}\int_{\Omega}tL(u(t,x))dxdt\leq\lim\inf\, \int_{0}^{T}\int_{\Omega }tL(u_{m}(t,x))dxdt, \end{array}$$

which follows from the inequality

$\begin{matrix} L (u_{m} (t, x)) \geq - C_{2}, \end{matrix}$ $$\begin{array}{} \displaystyle L(u_{m}(t,x))\geq-C_{2}, \end{array}$$

the convergence u_m(t, x) → u(t, x), for a.a. (t, x), and Lebesgue-Fatou’s lemma [35].

Bearing also in mind that

$\begin{matrix} {(u_{m} (T))}^{2} \to {(u (T))}^{2}, \\ \int_{0}^{T} t {(u (t))}_{V}^{2} d t \leq lim inf \int_{0}^{T} t {(u_{m} (t))}_{V}^{2} d t, \\ \int_{0}^{T} t {(u (t))}_{L^{p} (Ω)}^{p} d t s \leq lim inf \int_{0}^{T} t {(u_{m} (t))}_{L^{p} (Ω)}^{p} d t, \end{matrix}$ $$\begin{array}{} \displaystyle \qquad\qquad\,\,\left\Vert u_{m}\left( T\right) \right\Vert ^{2} \rightarrow\left\Vert u\left( T\right) \right\Vert ^{2},\\\displaystyle\qquad \int_{0}^{T}t\left\Vert u(t)\right\Vert _{V}^{2}dt \leq\lim\inf\, \int _{0}^{T}t\left\Vert u_{m}(t)\right\Vert _{V}^{2}dt,\\\displaystyle ~\int_{0}^{T}t\left\Vert u\left( t\right) \right\Vert _{L^{p}(\Omega)} ^{p}dts \leq\lim\inf\, \int_{0}^{T}t\left\Vert u_{m}\left( t\right) \right\Vert _{L^{p}(\Omega)}^{p}dt, \end{array}$$

we obtain readily that each term in the the left-hand side of (32) converges to the corresponding term in the left-hand side of (33).

Therefore,

$\begin{matrix} \int_{0}^{T} t {(u_{m} (t))}_{L^{p} (Ω)}^{p} d t \to \int_{0}^{T} t {(u (t))}_{L^{p} (Ω)}^{p} d t, \\ \int_{0}^{T} t {(u_{m} (t))}_{V}^{2} d t \to \int_{0}^{T} t {(u_{m} (t))}_{V}^{2} d t, \end{matrix}$ $$\begin{array}{} \displaystyle \int_{0}^{T}t\left\Vert u_{m}\left( t\right) \right\Vert _{L^{p}(\Omega )}^{p}dt\rightarrow\int_{0}^{T}t\left\Vert u\left( t\right) \right\Vert _{L^{p}(\Omega)}^{p}dt,\\ \displaystyle\quad\,\int_{0}^{T}t\left\Vert u_{m}(t)\right\Vert _{V}^{2}dt\rightarrow\int_{0} ^{T}t\left\Vert u_{m}(t)\right\Vert _{V}^{2}dt, \end{array}$$

and then for any 0 < r < T we get

$\begin{matrix} u_{m} \to u strongly in L^{2} (r, T; V) \cap L^{p} (r, T; L^{p} (Ω)), \end{matrix}$ $$\begin{array}{} \displaystyle u_{m}\rightarrow u\, \text{strongly in }L^{2}(r,T;V)\cap L^{p}(r,T;L^{p} (\Omega)), \end{array}$$

$\begin{matrix} u_{m} (t) \to u (t) in V \cap L^{p} (Ω) for a.a. t \in (0, T) . \end{matrix}$ $$\begin{array}{} \displaystyle u_{m}(t)\rightarrow u\left( t\right) \text{ in }V\cap L^{p}(\Omega)\text{ for a.a. }t\in\left( 0,T\right) . \end{array}$$

Applying then the dominated convergence theorem we deduce that F(u_m(t)) → F(u(t)) in L¹(Ω) for a.a. t ∈ (0, T). On top of that, in view of u_m → u in C([0, T], H), for any t ∈ [0, T] it follows that F(u_m(t, x)) → F(u(t, x)) for a.a. x ∈ Ω. Hence, by the inequality

$\begin{matrix} F (u_{m} (t, x)) \geq - D_{2} if p > 2, \\ F (u_{m} (t, x)) \geq δ {(u_{m} (t, x))}^{2} - D_{2} if p = 2, \end{matrix}$ $$\begin{array}{} \displaystyle F(u_{m}(t,x)) \geq-D_{2}\text{ if }p \gt 2,\\ F(u_{m}(t,x)) \geq\delta\left\vert u_{m}(t,x)\right\vert ^{2}-D_{2}\text{ if }p=2, \end{array}$$

and Lebesgue-Fatou’s lemma [35], we have

$\begin{matrix} \int_{Ω} F (u (t, x)) d x \leq lim inf \int_{Ω} F (u_{m} (t, x)) d x . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\Omega}F(u(t,x))dx\leq\lim\inf\, \int_{\Omega}F(u_{m}(t,x))dx. \end{array}$$

We know also from u_m(t) → u(t) weakly in V that

$\begin{matrix} {(u (t))}_{V} \leq lim inf {(u_{m} (t))}_{V} for all t \in [0, T] . \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert u\left( t\right) \right\Vert _{V}\leq\lim\inf\, \left\Vert u_{m}(t)\right\Vert _{V}\text{ for all }t\in\lbrack0,T]. \end{array}$$

Thus,

$\begin{matrix} E (u (t)) \leq lim inf E (u_{m} (t)) for all t \in [0, T] . \end{matrix}$ $$\begin{array}{} \displaystyle E(u(t))\leq\lim\inf\, E(u_{m}(t))\text{ for all }t\in\lbrack0,T]. \end{array}$$

Passing to the limit in (21) we obtain (29).

For any u₀ ∈ V ∩ L^p(Ω) let 𝓡(u₀) be the set of strong solutions to problem (14) which satisfy the energy inequality (29) for all t ≥ s and a.a. s > 0. This set is non-empty for every initial datum u₀ ∈ V ∩ L^p(Ω) due to Lemma 5. Denoting X = V ∩ L^p(Ω) we define now the multivalued mapping G : ℝ⁺ × X → P(X) by

$\begin{matrix} G (t, u_{0}) = {y : y = u (t) \in R (u_{0})} . \end{matrix}$ $$\begin{array}{} \displaystyle G(t,u_{0})=\{y:y=u(t)\in\mathscr{R}(u_{0})\}. \end{array}$$

It is straightforward to check that G(0, x) = x and G(t + s, x) ⊂ G(t, G(s, x)) for any x ∈ X and t, s ∈ ℝ⁺, that is, G is a multivalued semiflow.

In the sequel, $\begin{matrix} X_{w} = H_{0 w}^{1} (Ω) \cap L_{w}^{p} (Ω) . \end{matrix}$ $\begin{array}{} \displaystyle X_{w}=H_{0w}^{1}\left( \Omega\right) \cap L_{w}^{p}(\Omega). \end{array}$

Definition 5

The set 𝓐 is called a weak global attractor for the multivalued semiflow G if the following properties hold:

𝓐 is bounded in X and compact in X_w.

𝓐 is weakly attracting, that is, for any bounded set B in X and any neighborhood 𝓞 of 𝓐 in X_w there exists T(𝓞) such that

$\begin{matrix} G (t, B) \subset O for all t \geq T . \end{matrix}$ $$\begin{array}{} \displaystyle G(t,B)\subset\mathscr{O}\text{ for all }t\geq T. \end{array}$$

𝓐 is negatively semi-invariant, that is,

$\begin{matrix} A \subset G (t, A) for any t \geq 0. \end{matrix}$ $$\begin{array}{} \displaystyle \mathscr{A\subset}G(t,\mathscr{A)}\text{ for any }t\geq0. \end{array}$$

It follows from this definition that if K is a weakly closed set which is weakly attracting, then for any neighborhood 𝓞 of K in X_w there exists T(𝓞) such that

$\begin{matrix} A \subset G (t, A) \subset O for all t \geq T . \end{matrix}$ $$\begin{array}{} \displaystyle \mathscr{A\subset}G(t,\mathscr{A)\subset O}\text{ for all }t\geq T. \end{array}$$

Hence, 𝓐 ⊂ 𝓚 as X_w is a Hausdorff topological space. This means that 𝓐 is the minimal weakly closed set which is weakly attracting.

We recall that the set B₀ is said to be absorbing if for any bounded set B in X there exists T(B) such that

$\begin{matrix} G (t, B) \subset B_{0} for any t \geq T . \end{matrix}$ $$\begin{array}{} \displaystyle G(t,B)\subset B_{0}\text{ for any }t\geq T. \end{array}$$

Lemma 6

The multivalued semiflowGpossesses a bounded absorbing set $\begin{matrix} B_{0} = {v \in X : {(v)}_{V}^{2} + {(v)}_{L^{p} (Ω)}^{p} \leq R}, \end{matrix}$ $\begin{array}{} \displaystyle B_{0}=\{v\in X:\left\Vert v\right\Vert _{V}^{2}+\left\Vert v\right\Vert _{L^{p}(\Omega)}^{p}\leq R\}, \end{array}$whereRis a positive constant.

Remark 1

The set B₀ is weakly closed.

Proof

The function E(u(t)) satisfies (29) and multiplying the equation in (14) by u we have

$\begin{matrix} \frac{1}{2} \frac{d}{d t} {(u)}^{2} + {(u)}_{V}^{2} + α {(u)}_{L^{p} (Ω)}^{p} \leq C_{2} + \frac{1}{2 λ_{1}} {(h)}^{2} + \frac{λ_{1}}{2} {(u)}^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2}\frac{d}{dt}\left\Vert u\right\Vert ^{2}+\left\Vert u\right\Vert _{V}^{2}+\alpha\left\Vert u\right\Vert _{L^{p}(\Omega)}^{p}\leq C_{2}+\frac {1}{2\lambda_{1}}\left\Vert h\right\Vert ^{2}+\frac{\lambda_{1}}{2}\left\Vert u\right\Vert ^{2}. \end{array}$$

Therefore,

$\begin{matrix} {(u (t))}^{2} \leq e^{- λ_{1} t} {(u (0))}^{2} + K_{1}, for all t \geq 0, \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert u(t)\right\Vert ^{2}\leq e^{-\lambda_{1}t}\left\Vert u\left( 0\right) \right\Vert ^{2}+K_{1},\text{ for all }t\geq0, \end{array}$$

where $\begin{matrix} K_{1} = \frac{2 C_{2}}{λ_{1}} + \frac{1}{λ_{1}^{2}} {(h)}^{2}, \end{matrix}$ $\begin{array}{} \displaystyle K_{1}=\frac{2C_{2}}{\lambda_{1}}+\frac{1}{\lambda_{1}^{2}}\left\Vert h\right\Vert ^{2}, \end{array}$ and

$\begin{matrix} \int_{t}^{t + r} ({(u)}_{V}^{2} + {(u)}_{L^{p} (Ω)}^{p}) d t \leq K_{2} ({(u (t))}^{2} + r) \\ \leq K_{3} (e^{- λ_{1} t} {(u (0))}^{2} + 1 + r), \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t}^{t+r}\left( \left\Vert u\right\Vert _{V}^{2}+\left\Vert u\right\Vert _{L^{p}(\Omega)}^{p}\right) dt \leq K_{2}(\left\Vert u(t)\right\Vert ^{2}+r)\\ \displaystyle \qquad\qquad\qquad\qquad\qquad\quad~ \leq K_{3}(e^{-\lambda_{1}t}\left\Vert u\left( 0\right) \right\Vert ^{2}+1+r), \end{array}$$

for some positive constants K₂, K₃ and any r > 0, t ≥ 0.

Integrating in (29) over (t, t + r) we get

$\begin{matrix} E (t + r) \leq \frac{K_{3} (e^{- λ_{1} t} {(u (0))}^{2} + 1 + r)}{r}, for all r > 0, t \geq 0. \end{matrix}$ $$\begin{array}{} \displaystyle E(t+r)\leq\frac{K_{3}(e^{-\lambda_{1}t}\left\Vert u\left( 0\right) \right\Vert ^{2}+1+r)}{r}\text{, for all }r \gt 0,t\geq0. \end{array}$$

Taking into account that

$\begin{matrix} E (u (t)) = {(u (t))}_{V}^{2} + (F (u (t)), 1) - 2 (h, u (t)) \\ \geq {(u (t))}_{V}^{2} + δ {(u (t))}_{L^{p} (Ω)}^{p} - {(h)}^{2} - {(u (t))}^{2} - D_{2}, \end{matrix}$ $$\begin{array}{} \displaystyle E(u(t)) =\left\Vert u\left( t\right) \right\Vert _{V}^{2}+\left( F(u(t)),1\right) -2\left( h,u\left( t\right) \right) \\ \displaystyle\qquad\quad~ \geq\left\Vert u\left( t\right) \right\Vert _{V}^{2}+\delta\left\Vert u(t)\right\Vert _{L^{p}(\Omega)}^{p}-\left\Vert h\right\Vert ^{2}-\left\Vert u\left( t\right) \right\Vert ^{2}-D_{2}, \end{array}$$

it follows easily from the previous estimates the existence of a constant R > 0 such that the bounded set

$\begin{matrix} B_{0} = {v \in X : {(v)}_{V}^{2} + {(v)}_{L^{p} (Ω)}^{p} \leq R} \end{matrix}$ $$\begin{array}{} \displaystyle B_{0}=\{v\in X:\left\Vert v\right\Vert _{V}^{2}+\left\Vert v\right\Vert _{L^{p}(\Omega)}^{p}\leq R\} \end{array}$$

is absorbing for G.

The theory of global attractors in topological spaces for multivalued semiflows and processes has been developed for example in [1, 15, 21]. However, since the abstract conditions imposed in those papers are not met in our particular situation, we will prove the existence of the weak attractor from scratch.

Theorem 7

The multivalued semiflowGpossesses a weak global attractor 𝓐.

Proof

Take an arbitrary bounded set B. We recall that the ω − limit set of B is given by

$\begin{matrix} ω (B) = \cap_{s \geq 0} c l_{X_{w}} \cup_{t \geq s} G (t, B) . \end{matrix}$ $$\begin{array}{} \displaystyle \omega(B)=\cap_{s\geq0}cl_{X_{w}}\cup_{t\geq s}G(t,B). \end{array}$$

Since the space X_w satisfies the first axiom of countability, an equivalent definition for ω(B) is the following

$\begin{matrix} ω (B) = (\begin{matrix} y \in X : there exist sequences y_{n} \in G (t_{n}, x_{n}), t_{n} \to + \infty, x_{n} \in X \\ such that y_{n} \to y in X_{w} \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle \omega(B)=\left\{ \begin{array} [c]{c} y\in X:\text{ there exist sequences }y_{n}\in G(t_{n},x_{n})\text{, } t_{n}\rightarrow+\infty\text{, }x_{n}\in X\\ \text{such that }y_{n}\rightarrow y\text{ in }X_{w} \end{array} \right\} . \end{array}$$

Therefore, as X is a reflexive Banach space and by Lemma 6 the set ∪_{t≥ T(B)}G(t, B) is bounded for some T(B), any sequence y_n ∈ G(t_n, B) with t_n → +∞ has a weakly convergent subsequence. Thus, ω(B) is non-empty. It is obvious that ω(B) belongs to the absorbing set B₀, so it is bounded in X, and that it is weakly closed, so it is compact in X_w.

Further, let us check that ω(B) weakly attracts B. Assuming the opposite, there exists a neighborhood 𝓞 of ω(B) and a sequence y_n ∈ G(t_n, B), where t_n → +∞, such that y_n ∉ 𝓞. But this leads to a contradiction as from {y_n} we can extract a converging subsequence whose limit belongs to ω(B).

It remains to prove that ω(B) is negatively semi-invariant.

First of all, let us consider a sequence of strong solutions uⁿ(⋅) ∈ 𝓡( $\begin{matrix} u_{0}^{n} \end{matrix}$ $\begin{array}{} \displaystyle u_{0}^{n} \end{array}$ ) such that uⁿ(t) ∈ B₀ for any t ≥ 0. In view of well-known results (see [20, Lemma 15] and [16, Theorem 3.11]), there exists a weak solution u(⋅) to problem (14) and a subsequence u^n_k(⋅) such that u^n_k → u strongly in C([0, T], H), for all T > 0 (among other convergences). But uⁿ(t) are uniformly bounded in X, so

$\begin{matrix} u^{n_{k}} \to u weakly star in L^{\infty} (0, T; V \cap L^{p} (Ω)) for all T > 0, \\ u^{n_{k}} (t) \to u (t) in X_{w} for any t \geq 0. \end{matrix}$ $$\begin{array}{} \displaystyle u^{n_{k}}\rightarrow u\text{ weakly star in }L^{\infty}(0,T;V\cap L^{p} (\Omega))\text{ for all }T \gt 0,\\\displaystyle\qquad\qquad\quad u^{n_{k}}(t)\rightarrow u\left( t\right) \text{ in }X_{w}\text{ for any }t\geq0. \end{array}$$

Also, inequalities (29), (6) and E(uⁿ(s)) ≤ C, for any s ≥ 0 and n, imply that $\begin{matrix} \frac{d u^{n}}{d t} \end{matrix}$ $\begin{array}{} \displaystyle \dfrac{du^{n}}{dt} \end{array}$ are bounded in L²(0, T; H) and then

$\begin{matrix} \frac{d u^{n_{k}}}{d t} \to \frac{d u}{d t} weakly in L^{2} (0, T; H) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{du^{n_{k}}}{dt}\rightarrow\frac{du}{dt}\text{ weakly in }L^{2}(0,T;H). \end{array}$$

Therefore, u(⋅) is in fact a strong solution to problem (14). On top of that, arguing in the same way as in Lemma 5 we obtain that

$\begin{matrix} u^{n_{k}} \to u strongly in L^{2} (r, T; V) \cap L^{p} (r, T; L^{p} (Ω)), for any 0 < r < T, \end{matrix}$ $$\begin{array}{} \displaystyle u^{n_{k}}\rightarrow u\, \text{strongly in }L^{2}(r,T;V)\cap L^{p} (r,T;L^{p}(\Omega)),\text{ for any }0 \lt r \lt T, \end{array}$$

and also that (29) is satisfied for all t ≥ s and a.a. s > 0. Thereby, u(⋅) ∈ 𝓡(u₀).

Consider now an arbitrary element y ∈ ω(B) and t > 0. Then there is a sequence y_n ∈ G(t_n, x_n) with t_n → +∞, x_n ∈ B such that y_n → y in X_w. In addition, we take N(B) for which G(s, B) ⊂ B₀ if s ≥ t_n − t and n ≥ N. Since G(t_n, x_n) ⊂ G(t, G(t_n − t, x_n)), there are z_n ∈ G(t_n − t, x_n), uⁿ(⋅) ∈ 𝓡(z_n) satisfying that uⁿ(s) ∈ B₀, for any s ≥ 0, and y_n = uⁿ(t). Passing to a subsequence z_n converges in the space X_w to some z ∈ ω(B). In light of the previous arguments there exists u(⋅) ∈ 𝓡(z) such that u(t) = y. Thus, y ∈ G(t, z) ⊂ G(t, ω(B)), which proves that ω(B) is negatively semi-invariant.

Finally, let 𝓐 = ω(B₀). Obviously, 𝓐 is bounded in X, compact in X_w and negatively semi-invariant. The weakly attracting property follows from the chain of inclusions

$\begin{matrix} G (t, B) \subset G (t - T (B), G (T (B), B)) \subset G (t - T (B), B_{0}) \end{matrix}$ $$\begin{array}{} \displaystyle G(t,B)\subset G(t-T(B),G(T(B),B))\subset G(t-T(B),B_{0}) \end{array}$$

and the fact that 𝓐 weakly attracts B₀.

eISSN:: 2444-8656
Language:: English

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On Lr-regularity of global attractors generated by strong solutions of reaction-diffusion equations

Published Online: Sep 16, 2016

Page range: 375 - 390

Received: Jul 28, 2016

Accepted: Sep 16, 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00033

Keywordsreaction-diffusion equation, setvalued dynamical system, global attractor, multivalued semiflow

© 2016 José Valero, published by Sciendo

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

On L^r-regularity of global attractors generated by strong solutions of reaction-diffusion equations

Keywords
reaction-diffusion equation, setvalued dynamical system, global attractor, multivalued semiflow