H1) Concerning the nonlinearity, we assume that f β C1(β, β) there exist positive constants c, ΞΌ0, ΞΌ1, k and p > 2 N β€
$\begin{array}{}
\frac{2p}{p-2}
\end{array}$
such that
$$\begin{array}{}
\displaystyle
-c - \mu_0 \vert u \vert^p \leq f(u)u\leq c - \mu_1 \vert u \vert^p \; \forall u \in \mathbb{R},
\end{array}$$
$$\begin{array}{}
\displaystyle
\left(f(u)-f(v)\right)(u-v) \leq k (u-v)^2 \; \forall u, v \in \mathbb{R}.
\end{array}
$$
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 Ξ½ > 0 is the kinematic viscosity, u is the velocity field of the fluid, p the pressure, Ο β β the initial time, u0 the initial velocity field, f a nondelayed external force field, g another external force with delay and Ο the initial condition in (βh,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:
In [3], J.Garcia-Luengo and P.Marin-Rubio treated the following reaction-diffusion equation with non-autonomous force in Hβ1 and delays under measurability conditions on the driving delay term:
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 π-attractor. Then we can prove the existence of pullback π-attractor for the nonautonomous problem with delay.
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 β ββand all T > Ο, the sequence {um} is bounded in
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 Ο1, Ο2. Denoting that w = u1 β 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, π«(X) be the class of nonempty subsets of X, and suppose π is a nonempty class of parameterized sets DΜ = {D(t) : t β β}β π«(X).
Definition 2
A two parameter family of mappings U(t,Ο) : X β X t β₯ Ο, Ο β β, is called to be a process if
S(Ο,Ο)x = {x},β Ο β β, x β Y;
S(t,s)S(s,Ο)x = S(t,Ο)x, β t β₯ s β₯ Ο, Ο β β, x β X.
Definition 3
A family of bounded sets BΜ = {B(t) : t β β}β π is called pullback π-absorbing for the processS(t,Ο)} if for any t β β and for any DΜ β π, there exists Ο0(t,DΜ) β€ t such that
The process S(t,Ο) is said to be pullback π-asymptotically compact if for all t β β, all DΜ β π, any sequence Ο n β -β, and any sequence xn β D(Οn), the sequence {S(t,Οn)xn} is relatively compact in X.
Definition 5
A family Γ = {A(t) : t β β}β π«(X) is said to be a pullback 7 π-attractor forS(t,Ο)} if
A(t) is compact for all t β β;
Γ is invariant; i.e., S(t,Ο)A(Ο) = A(t), for all t β₯ Ο;
If {C(t) : t β β} is another family of closed attracting sets then A(t) β C(t), for all t β β.
Theorem 2
Let us suppose that the process {S(t,Ο)} is pullback π-asymptotically compact, and BΜ = {B(t) : t β β}β π is a family of pullback π-absorbing sets for {S(t,Ο)}. Then there exists a pullback π-attractor {A(t) : t β β} such that
To check the continuity of the process, we need the following lemma.
Lemma 1
Let (u0,Ο), (v0,Ο)β 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 Ξ½: =
$\begin{array}{}
2(\frac{1}{2}+k+\frac{C_b}{2}-\lambda_1)>0\,,
\end{array}$such that
By this last estimate we finished the proof of this lemma. βΌ
Theorem 3
Under the previous assumptions, the mapping S(.,.) defined in(3.1), is a continuous process for all Ο β€ 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,Ο), (v0,Ο)β H and u, v are their corresponding solutions. Firstly, if we take t β₯ Ο + r, it follows from (3.4)
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.
Concerning the second part, we observe that {j(B(t)), t β β} is a family of pullback π-absorbing sets for the process S. On the other hand, since
$$\begin{array}{}
\displaystyle
\Vert \varphi\Vert_{L^2([-r,0];L^2(\Omega))}^2 \leq r \Vert \varphi\Vert_{C([-r,0];L^2(\Omega))}^2 \,,
\end{array}$$
Since Ξ± in lemma (3) is small enough, we can choose a positive constant Ξ±β sufficiently small with Ξ± < Ξ±* <
$\begin{array}{}
\displaystyle
\min \left\{ 2 \frac{\lambda_1 -1}{\lambda_1}\,,\, 2\mu_1 \right\}\,,
\end{array}$ 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 π 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) β
$\begin{array}{}
\displaystyle
\mathbf{\overline{B}}_{C([-r,0];H^1_0(\Omega))}(0,\rho(t))\,,
\end{array}$
for some Ο β π‘, 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 Ο(t). Let
for all Ο β€ Ο0(DΜ, 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 π-absorbing for the mapping U(t, Ο).
The proof of the proposition is completed. βΌ
Existence of pullback D-attractor
To prove the existence of pullback π-attractor, we need to prove the following lemma.
Lemma 4
Assume that conditions oflemma (3)are satisfied. Then the process {S(t, Ο)} corresponding to(1.1)is pullback π-asymptotically compact.
Proof
Let t β β, DΜ β π, a sequences Οn βnβ+β β β and (u0, n, Οn β D(Οn), be fixed. We have to check that the sequence
By Proposition 1 and Lemma 4, we proved that the process S(t, Ο) has a pullback π-absorbing set and it is pullback π-asymptotically compact, then by Theorem 2 we can deduce the following result.