where Ω ⊂ ℝℕ is a bounded domain with smooth boundary ∂ Ω, τ ∈ ℝ, u0 ∈ L2(Ω) is the initial condition in τ and φ ∈ L2([–r,0];L2(Ω)) is also the initial condition in [τ–r,τ],r > 0 is the length of the delay effect. For the rest we assume following assumptions conditions:
H1) Concerning the nonlinearity, we assume that f ∈ C1(ℝ, ℝ) there exist positive constants c, μ0, μ1, k and p > 2 N ≤
$\begin{array}{}
\frac{2p}{p-2}
\end{array}$
such that
$$\begin{array}{}
\displaystyle
-c - \mu_0 \vert u \vert^p \leq f(u)u\leq c - \mu_1 \vert u \vert^p \; \forall u \in \mathbb{R},
\end{array}$$
$$\begin{array}{}
\displaystyle
\left(f(u)-f(v)\right)(u-v) \leq k (u-v)^2 \; \forall u, v \in \mathbb{R}.
\end{array}
$$
From (I)-(III), for T > τ and u ∈ L2([τ–r,T];L2(Ω)) the function ℝ∋ t ↦ b(t,ϕ) ∈ L2(Ω) is measurable and belongs to L∞((τ,T);L2(Ω)).
H3) The function g ∈
$\begin{array}{}
L^2_{loc}
\end{array}$
(ℝ; L2(Ω)) is an another nondelayed time-dependent external force.
For more details on differential equations with delay, we refer the reader to J. Wu [9] and J.K. Hale [5]. The purpose of this paper is to discuss the existence of pullback 𝒟-attractor in L2(Ω)× L2([–r,0];L2(Ω)) by using a priori estimates of solutions to the problem (1.1).
This work is motivated by the work of T. Caraballo and J. Real. [1], where they proved the existence of pullback attractors for the following 2D-Navier Stokes model with delays:
where ν > 0 is the kinematic viscosity, u is the velocity field of the fluid, p the pressure, τ ∈ ℝ the initial time, u0 the initial velocity field, f a nondelayed external force field, g another external force with delay and ϕ the initial condition in (–h,0), where h is a fixed positive number.
On the other hand, the problem (1.1) without critical nonlinearity was treated by J. Li and J. Huang in [6], where they proved the existence of uniform attractor for the following non-autonomous parabolic equation with delays:
Here Ω is a bounded domain in ℝn0 with smooth boundary, b ≥ 0, A is a densely-defined self-adjoint positive linear operator with domain D(A)⊂ L2(Ω) and with compact resolvent, F is the nonlinear term which is locally Lipschitz continuous for the initial condition, g is an external force.
In [3], J.Garcia-Luengo and P.Marin-Rubio treated the following reaction-diffusion equation with non-autonomous force in H–1 and delays under measurability conditions on the driving delay term:
where τ ∈ ℝ, f ∈ C(ℝ) the nonlinear term with critical exponent, g is an external force with delay, k ∈
$\begin{array}{}
L^{2}_{loc}
\end{array}$
(ℝ;H–1(Ω)) a time-dependent force, ϕ the initial condition and h the lenght of the delay effect. In this work, the authors checked the existence of pullback 𝒟-attractor in C([–h,0]; L2(Ω)).
This paper is organized as follows. In section 2, we will prove the existence of weak solutions to the problem (1.1) by using the Faedo-Galerkin approximations, as well as the uniqueness and the continuous dependence of solution with respect to initial conditions. In section 3, we recall some definitions and abstract results on pullback 𝒟-attractor. Then we can prove the existence of pullback 𝒟-attractor for the nonautonomous problem with delay.
Existence and uniqueness of solution
First we give the concept of the solution.
Definition 1
A weak solution of(1.1)is a function u ∈ L2([τ–r,T];L2(Ω)) such that for all T > τ we have
for all test functions v ∈ L2([τ, T];
$\begin{array}{}
\displaystyle
H^{1}_{0}
\end{array}$
(Ω)) and v′∈ L2([τ, T]; H–1(Ω)) such that v(T) = 0.
Theorem 1
Assume that g ∈
$\begin{array}{}
\displaystyle
L^{2}_{log}
\end{array}$
(ℝ;L2(Ω)), b and f satisfy (I)-(IV) and(1.2)-(1.5)respectively and ifλ1 > 1 + Cb/2, Then for all T > τ and all (u0,φ) in L2(Ω)× L2([–r,0];L2(Ω)), there exists a unique weak solution u to the problem(1.1).
Proof
Let us consider {ek}k ≥ 1, the complete basis of
$\begin{array}{}
\displaystyle
H^{1}_{0}
\end{array}$
(Ω) which is given by the orthonormal eigenfunctions of Δ in L2(Ω). We consider
for all k = 1 … m. Where γk,m(t) = 〈 um(t), ek 〉 denote the Fourier coefficients; such that γm,k ∈ C1((τ, T); ℝ) ∩ L2((τ–r, T), ℝ),
$\begin{array}{}
\gamma'_{k,m}
\end{array}$
(t) is absolutely continuous, and Pmu(t) =
$\begin{array}{}
\sum^{m}_{k=1}
\end{array}$
〈u,ek〉 ek is the orthogonal projection of L2(Ω) and
$\begin{array}{}
\displaystyle
H^{1}_{0}
\end{array}$
(Ω) in Vm = span{e1, …, em}.
It is well-known that the above finite-dimensional delayed system is well-posed (e.g. cf. [2]), at least locally. We will provide a priori estimates for the Faedo-Galerkin approximate solutions.
For all m ∈ ℕ∗and all T > τ, the sequence {um} is bounded in
From (2.1) we deduce that the term
$\begin{array}{}
\int_{\tau}^{t} \Vert u^m(s) \Vert_{L^p(\Omega)}^p
\end{array}$ds is bounded, so by this last estimate we conclude that {f(um)} is bounded in Lq((τ, T);Lq(Ω)), for all T > τ.
$\begin{array}{}
\left\{\frac{\partial }{\partial t}u^m\right\}
\end{array}$is bounded in L2((τ,T);L2(Ω)).
Now, multiplying (1.1) by
$\begin{array}{}
\frac{\partial u^m}{\partial t}
\end{array}$
and integrating over Ω, one has
By the Aubin-Lions lemma of compactness, we conclude that um → u strongly in L2((τ,T);L2(Ω)). Thus um → u a.e [τ,T]× Ω.
Since f is continuous, we deduce that f(um) → f(u) a.e [τ,T]× Ω. So from (2.3) and (lemma 1.3 in [7], p.12) we can identify σ′ with f(u).
To prove that u(τ) = u0, we put v ∈ C1((τ,T);
$\begin{array}{}
\displaystyle
H^{1}_{0}
\end{array}$
(Ω)) such that v(T) = 0 and we note from (1.1) that
Since um(τ)→ u0 and um(τ + θ)→ φ(θ), the estimate (2.10) shows that um → u uniformly in C([τ,T];L2(Ω)).
Finally, we prove the uniqueness and continuous dependence of the solution. Let u1; u2 be two solutions of problem (1.1) with the initial conditions u0,1, u0,2 and φ1, φ2. Denoting that w = u1 – u2 and repeating the argument as in the proof of (2.10), we find
First, we give some basic definitions and an abstract result on the existence of pullback attractors, which we need to obtain our results (we refer the reader to [2,3,4,8]). Let (X,d) be a complete metric space, 𝒫(X) be the class of nonempty subsets of X, and suppose 𝒟 is a nonempty class of parameterized sets D̂ = {D(t) : t ∈ ℝ}⊂ 𝒫(X).
Definition 2
A two parameter family of mappings U(t,τ) : X → X t ≥ τ, τ ∈ ℝ, is called to be a process if
S(τ,τ)x = {x},∀ τ ∈ ℝ, x ∈ Y;
S(t,s)S(s,τ)x = S(t,τ)x, ∀ t ≥ s ≥ τ, τ ∈ ℝ, x ∈ X.
Definition 3
A family of bounded sets B̂ = {B(t) : t ∈ ℝ}∈ 𝒟 is called pullback 𝒟-absorbing for the processS(t,τ)} if for any t ∈ ℝ and for any D̂ ∈ 𝒟, there exists τ0(t,D̂) ≤ t such that
The process S(t,τ) is said to be pullback 𝒟-asymptotically compact if for all t ∈ ℝ, all D̂ ∈ 𝒟, any sequence τ n → -∞, and any sequence xn ∈ D(τn), the sequence {S(t,τn)xn} is relatively compact in X.
Definition 5
A family  = {A(t) : t ∈ ℝ}⊂ 𝒫(X) is said to be a pullback 7 𝒟-attractor forS(t,τ)} if
A(t) is compact for all t ∈ ℝ;
 is invariant; i.e., S(t,τ)A(τ) = A(t), for all t ≥ τ;
If {C(t) : t ∈ ℝ} is another family of closed attracting sets then A(t) ⊂ C(t), for all t ∈ ℝ.
Theorem 2
Let us suppose that the process {S(t,τ)} is pullback 𝒟-asymptotically compact, and B̂ = {B(t) : t ∈ ℝ}∈ 𝒟 is a family of pullback 𝒟-absorbing sets for {S(t,τ)}. Then there exists a pullback 𝒟-attractor {A(t) : t ∈ ℝ} such that
with (u0,φ)∈ H. To this aim, We consider g ∈
$\begin{array}{}
L^{2}_{loc}
\end{array}$
(ℝ;L2(Ω)), b: ℝ × L2([–r,0];L2(Ω)) → L2(Ω) with the hypotheses (I)-(IV) and f ∈ C1(ℝ;ℝ) verifying (1.2)-(1.5). Then the family of mappings
To check the continuity of the process, we need the following lemma.
Lemma 1
Let (u0,φ), (v0,ϕ)∈ H be two couples of initial conditions for the problem(1.1)and u, v be the corresponding solutions to(1.1). Then there exists a positive constant ν: =
$\begin{array}{}
2(\frac{1}{2}+k+\frac{C_b}{2}-\lambda_1)>0\,,
\end{array}$such that
By this last estimate we finished the proof of this lemma. ◼
Theorem 3
Under the previous assumptions, the mapping S(.,.) defined in(3.1), is a continuous process for all τ ≤ t.
Proof
The proof of this theorem is as the proof of Theorem 9 in [1]. The uniqueness of the solutions implies that S(.,.) is a process. For the continuity of S(.,.), we use the previous lemma. We consider (u0,φ), (v0,ϕ)∈ H and u, v are their corresponding solutions. Firstly, if we take t ≥ τ + r, it follows from (3.4)
Hence, by this last estimate and (3.3) we deduce the continuity of S(t,τ).◼
Existence of pullback D-absorbing set in C([–r,0]; L2(Ω)) and H
Firstly, we need to the following lemma, it relates the absorption properties for the mappings with those of process S in the fact that, proving those for U yields to similar properties for S.
Lemma 2
Assume that the family of bounded sets {B(t): t ∈ ℝ} in the space C([–r,0]; L2(Ω)) is pullback 𝒟-absorbing for the mapping U(.,.). Then the family of bounded sets {j(B(t)): t ∈ ℝ} in L2(Ω) × C([–r,0]; L2(Ω)) is pullback 𝒟-absorbing for the process S(.,.).
Proof
Let {D(t): t ∈ ℝ} be a family bounded sets in H, so there exists T > r such that
By 𝓓 we denote the class of all families D̂ = {D(t) : t ∈ ℝ} ⊂ 𝓟(C([–r, 0];L2(Ω))) such that D(t) ⊂ BC([–r, 0];L2(Ω))(0,ρ(t)), for some ρ ∈ 𝓡, where we denote by BC([–r, 0];L2(Ω))(0, ρ(t)) the closed ball in C([–r, 0];L2(Ω)) centered at 0 with radius ρ(t). Let
Since α in lemma (3) is small enough, we can choose a positive constant α∗ sufficiently small with α < α* <
$\begin{array}{}
\displaystyle
\min \left\{ 2 \frac{\lambda_1 -1}{\lambda_1}\,,\, 2\mu_1 \right\}\,,
\end{array}$ such that
On the one hand, since
$\begin{array}{}
\displaystyle
H^1_0(\Omega) \subset L^2(\Omega)\, {\rm and}\, H^1_0(\Omega) \subset L^p(\Omega)\,,
\end{array}$we have
by 𝓓 we denote the class of all families
$\begin{array}{}
\displaystyle
\mathbf{\widehat{D}} = \{D(t) : t\in \mathbb{R} \} \subset \mathcal{P}(C([-r,0];H^1_0(\Omega)))
\end{array}$
such that D(t) ⊂
$\begin{array}{}
\displaystyle
\mathbf{\overline{B}}_{C([-r,0];H^1_0(\Omega))}(0,\rho(t))\,,
\end{array}$
for some ρ ∈ 𝓡, where we denote by
$\begin{array}{}
\displaystyle
\mathbf{\overline{B}}_{C([-r,0];H^1_0(\Omega))}(0,\rho(t))
\end{array}$
the closed ball in
$\begin{array}{}
\displaystyle
C([-r,0];H^1_0(\Omega))
\end{array}$
centered at 0 with radius ρ(t). Let
for all τ ≤ τ0(D̂, t), this means that
$\begin{array}{}
\displaystyle
B_2(t) = \overline{B}_{C([-r,0]; H^1_0(\Omega))}(0, R_2(t))
\end{array}$
is pullback 𝓓-absorbing for the mapping U(t, τ).
The proof of the proposition is completed. ◼
Existence of pullback D-attractor
To prove the existence of pullback 𝓓-attractor, we need to prove the following lemma.
Lemma 4
Assume that conditions oflemma (3)are satisfied. Then the process {S(t, τ)} corresponding to(1.1)is pullback 𝓓-asymptotically compact.
Proof
Let t ∈ ℝ, D̂ ∈ 𝓓, a sequences τn →n→+∞ – ∞ and (u0, n, φn ∈ D(τn), be fixed. We have to check that the sequence
is relatively compact in C([–r, 0];L2(Ω)). To this end, we use the Ascoli-Arzela theorem. In other words, we check
the equicontinuity property for the sequence
$\begin{array}{}
\displaystyle
\{u_t(.,\tau_n, (u^{0,n},\varphi^n))\}:= \{u_t^n(.)\}\,, \rm i.e.\,\, \forall \varepsilon \gt 0, \, \exists \delta \gt 0
\end{array}$
such that if
$\begin{array}{}
\displaystyle
\vert \theta_1 -\theta_2 \vert \leq \delta \,,\, \rm then\, \Vert u^n_t(\theta_1) - u^n_t(\theta_2) \Vert \leq \varepsilon\,,\, for\, all\, \,\theta_1 \gt \theta_2 \in [-r,0]\,
\end{array}$;
the uniform boundedness of
$\begin{array}{}
\displaystyle
\{u^n_t(\theta)\}\,,
\end{array}$
for all θ ∈ [–r, 0].
In order to prove (b), we consider un, u the corresponding solutions to (1.1), so by Lemma 1 we can deduce that
$\begin{array}{}
\displaystyle
\{u^n_t\}
\end{array}$
and {ut} are uniformly bounded in C([–r, 0]; L2(Ω)).
and this ensures the equicontinuity property in C([–r, 0]; L2(Ω)); i.e. the sequence {U(t, τn)(u0, n, φn)} is relatively compact in C([–r, 0]; L2(Ω)).
Since we have S(t, τn)(u0, n, φn) = j(U(t, τn)(u0, n, φn)), so {S(t, τn)(u0, n, φn)} is relatively compact in the space L2(Ω) × C([–r, 0]; L2(Ω)) and by the continuous injection of L2(Ω) × C([–r, 0]; L2(Ω)) in H, we deduce that {S(t, τn)(u0, n, φn)} is relatively compact in H. The proof of this lemma is completed. ◼
By Proposition 1 and Lemma 4, we proved that the process S(t, τ) has a pullback 𝓓-absorbing set and it is pullback 𝓓-asymptotically compact, then by Theorem 2 we can deduce the following result.
Theorem 4
The process {S(t, τ)} corresponding to(1.1)has a pullback 𝓓-attractor = {A(t) : t ∈ ℝ} in H. Furetheremore,  ⊂ L2(Ω) × C([–r, 0]; L2(Ω)).