Attractors for a nonautonomous reaction-diffusion equation with delay

$\begin{matrix} (\begin{matrix} \frac{\partial}{\partial t} u (t, x) - Δ u (t, x) = f (u (t, x)) + b (t, u_{t}) (x) + g (t, x) in (τ, \infty) \times Ω, \\ u = 0 on (τ, \infty) \times \partial Ω, \\ u (τ, x) = u^{0} (x), τ \in R and x \in Ω, \\ u (τ + θ, x) = φ (θ, x), θ \in [- r, 0] and x \in Ω, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{l} \frac{\partial}{\partial t} u(t,x) - \Delta u(t,x) = f(u(t,x)) + b(t,u_t)(x) + g(t,x)\; \mbox{in}\; (\tau, \infty) \times \Omega\,,\\ u= 0 \;\mbox{on}\; (\tau, \infty) \times \partial{\Omega}\,,\\ u(\tau,x)=u^0(x),\; \tau \in \mathbb{R}\; \mbox{and} \;x \in \Omega \,,\\ u(\tau+\theta,x)= \varphi(\theta,x),\; \theta \in [-r,0] \;\mbox{and}\; x \in \Omega\,, \end{array} \right. \end{array}$$(1.1)

where Ω ⊂ ℝ^ℕ 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 ≤ $\begin{matrix} \frac{2 p}{p - 2} \end{matrix}$ $\begin{array}{} \frac{2p}{p-2} \end{array}$ such that

$\begin{matrix} - c - μ_{0} | u |^{p} \leq f (u) u \leq c - μ_{1} | u |^{p} \forall u \in R, \end{matrix}$ $$\begin{array}{} \displaystyle -c - \mu_0 \vert u \vert^p \leq f(u)u\leq c - \mu_1 \vert u \vert^p \; \forall u \in \mathbb{R}, \end{array}$$(1.2)

$\begin{matrix} (f (u) - f (v)) (u - v) \leq k (u - v)^{2} \forall u, v \in R . \end{matrix}$ $$\begin{array}{} \displaystyle \left(f(u)-f(v)\right)(u-v) \leq k (u-v)^2 \; \forall u, v \in \mathbb{R}. \end{array} $$(1.3)

Let us denote by

$\begin{matrix} F (u) := \int_{0}^{u} f (s) d s . \end{matrix}$ $$\begin{array}{} \displaystyle F(u) := \int_{0}^{u} f(s) ds\,. \end{array}$$

From (1.2), there exist positive constants l,c′ $\begin{matrix} μ_{0}^{'}, μ_{1}^{'} \end{matrix}$ $\begin{array}{} \mu'_0, \mu'_1 \end{array}$ such that

$\begin{matrix} | f (u) | \leq l (| u |^{p - 1} + 1) \forall u \in R, \end{matrix}$ $$\begin{array}{} \displaystyle \vert f(u) \vert \leq l\left(\vert u \vert^{p-1} + 1\right)\; \forall u\in \mathbb{R}, \end{array}$$(1.4)

$\begin{matrix} - c^{'} - μ_{0}^{'} | u |^{p} \leq F (u) \leq c^{'} - μ_{1}^{'} | u |^{p} \forall u \in R . \end{matrix}$ $$\begin{array}{} \displaystyle -c' - \mu'_0 \vert u \vert^p \leq F(u)\leq c' - \mu'_1 \vert u \vert^p \; \forall u \in \mathbb{R}. \end{array}$$(1.5)

H₂) The operator b : ℝ × L²([–r,0];L²(Ω)) → L²(Ω) is a time-dependent external force with delay, such that

For all ϕ ∈ L²([–r,0]; L²(Ω)) the function ℝ∋ t ↦ b(t,ϕ) ∈ L²(Ω) is measurable;

b(t,0) = 0 for all t ∈ ℝ;

∃ L_b > 0 s.t ∀ t ∈ ℝ and ∀ ϕ₁, ϕ₂ ∈ L²([–r,0];L²(Ω));

$\begin{matrix} ∥ b (t, ϕ_{1}) - b (t, ϕ_{2}) ∥ \leq L_{b} ∥ ϕ_{1} - ϕ_{2} ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}; \end{matrix}$ $$\begin{array}{} \displaystyle \Vert b(t,\phi_1)-b(t,\phi_2)\Vert \leq L_b \Vert \phi_1-\phi_2 \Vert_{L^2([-r,0];L^2(\Omega))}\,; \end{array}$$(1.6)

∃ C_b > 0 s.t ∀ t ≥ τ, and ∀ u, v ∈ L²([τ-r, t]; L²(Ω));

$\begin{matrix} \int_{τ}^{t} ∥ b (s, u_{s}) - b (s, v_{s}) ∥^{2} d s \leq C_{b} \int_{τ - r}^{t} ∥ u (s) - v (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau}^{t}\Vert b(s,u_s)-b(s,v_s)\Vert^2 ds \leq C_b \int_{\tau-r}^{t}\Vert u(s)-v(s) \Vert^2 ds\,. \end{array}$$(1.7)

Remark 1

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 ∈ $\begin{matrix} L_{l o c}^{2} \end{matrix}$ $\begin{array}{} L^2_{loc} \end{array}$ (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:

$\begin{matrix} (\begin{matrix} \frac{\partial u}{\partial t} - ν Δ u + \sum_{i = 1}^{2} u_{i} \frac{\partial u}{\partial x_{i}} = f - \nabla p + g (t, u_{t}) in (τ, \infty) \times Ω, \\ d i v u = 0 in (τ, \infty) \times Ω, \\ u = 0 on (τ, \infty) \times \partial Ω, \\ u (τ, x) = u_{0} (x), x \in Ω, \\ u (t, x) = ϕ (t - τ, x), t \in (τ - h, τ) and x \in Ω, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{l} \frac{\partial u}{\partial t} - \nu\Delta u + \sum_{i=1}^{2} u_i \frac{\partial u}{\partial x_i}= f - \nabla p + g(t,u_t)\; \mbox{in}\; (\tau, \infty) \times \Omega\,,\\ div\; u= 0 \;\mbox{in}\; (\tau, \infty) \times \Omega\,,\\ u= 0 \;\mbox{on}\; (\tau, \infty) \times \partial\,{\Omega}\,,\\ u(\tau,x)=u_0(x), \;x \in \Omega \,,\\ u(t,x)= \phi(t-\tau,x),\; t \in (\tau-h,\tau) \;\mbox{and}\; x \in \Omega\,, \end{array} \right. \end{array}$$(1.8)

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:

$\begin{matrix} (\begin{matrix} \frac{\partial u (t, x)}{\partial t} + A u (t, x) + b u (t, x) = F (u_{t}) (x) + g (t, x) x in Ω, \\ u (τ, x) = u_{0} (x), u (τ + θ, x) = ϕ (θ, x), θ \in (- r, 0) . \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{l} \frac{\partial u(t,x)}{\partial t} + A u(t,x) + b u(t,x) = F(u_t)(x) + g(t,x)\; x\;\mbox{in}\; \Omega\,,\\ u(\tau,x)=u_0(x),\; u(\tau+\theta,x)= \phi(\theta,x),\; \theta \in (-r,0) \,. \end{array} \right. \end{array}$$(1.9)

Here Ω is a bounded domain in ℝ^n₀ 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:

$\begin{matrix} (\begin{matrix} \frac{\partial u}{\partial t} - Δ u = f (u) + g (t, u_{t}) + k (t) in (τ, \infty) \times Ω, \\ u = 0 on (τ, \infty) \times \partial Ω, \\ u (τ + s, x) = ϕ (s, x), s \in [- r, 0] and x \in Ω, \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{l} \frac{\partial u}{\partial t} - \Delta u = f(u) + g(t,u_t) + k(t)\; \mbox{in}\; (\tau, \infty) \times \Omega\,,\\ u= 0 \;\mbox{on}\; (\tau, \infty) \times \partial\,{\Omega}\,,\\ u(\tau+s,x)= \phi(s,x),\; s \in [-r,0] \;\mbox{and}\; x \in \Omega\,, \end{array} \right. \end{array}$$(1.10)

where τ ∈ ℝ, f ∈ C(ℝ) the nonlinear term with critical exponent, g is an external force with delay, k ∈ $\begin{matrix} L_{l o c}^{2} \end{matrix}$ $\begin{array}{} L^{2}_{loc} \end{array}$ (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

$\begin{matrix} u \in L^{2} ((τ, T); H_{0}^{1} (Ω)) \cap L^{p} ((τ, T); L^{p} (Ω)) \cap C ([τ, T]; L^{2} (Ω)) \end{matrix}$ $$\begin{array}{} \displaystyle u \in L^2((\tau,T);H^1_0(\Omega)) \cap L^p((\tau,T);L^p(\Omega)) \cap C([\tau,T]; L^2(\Omega)) \end{array}$$

and

$\begin{matrix} \frac{\partial u}{\partial t} \in L^{2} ([τ, T]; L^{2} (Ω)), \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial u}{\partial t} \in L^2([\tau, T]; L^2(\Omega))\,, \end{array}$$

with u(t) = φ(t–τ), for t ∈ [τ–r,τ], and it satisfies

$\begin{matrix} \int_{τ}^{T} - 〈 u, v^{'} 〉 + \int_{τ}^{T} \int_{Ω} \nabla u \nabla v = \int_{τ}^{T} \int_{Ω} f (u) v + \int_{τ}^{T} 〈 b (t, u_{t}), v 〉 \\ + \int_{τ}^{T} \int_{Ω} g v + 〈 u^{0}, v (τ) 〉, \end{matrix}$ $$\begin{array}{} \displaystyle \int^{T}_{\tau} - \langle u,v'\rangle + \int_{\tau}^{T}\int_{\Omega} \nabla u \nabla v = \int_{\tau}^{T}\int_{\Omega} f(u)v + \int_{\tau}^{T} \langle b(t,u_t), v\rangle \nonumber\\ \displaystyle \qquad \qquad\qquad\qquad\qquad\qquad~+ \int_{\tau}^{T}\int_{\Omega} gv + \langle u^0,v(\tau)\rangle \,, \end{array}$$

for all test functions v ∈ L²([τ, T]; $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H^{1}_{0} \end{array}$ (Ω)) and v′∈ L²([τ, T]; H^–1(Ω)) such that v(T) = 0.

Theorem 1

Assume that g ∈ $\begin{matrix} L_{l o g}^{2} \end{matrix}$ $\begin{array}{} \displaystyle L^{2}_{log} \end{array}$ (ℝ;L²(Ω)), b and f satisfy (I)-(IV) and(1.2)-(1.5)respectively and ifλ₁ > 1 + C_b/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 {e_k}_{k ≥ 1}, the complete basis of $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H^{1}_{0} \end{array}$ (Ω) which is given by the orthonormal eigenfunctions of Δ in L²(Ω). We consider

$\begin{matrix} u^{m} (t) = \sum_{k = 1}^{m} γ_{k, m} (t) e_{k}, m = 1, 2, \dots \end{matrix}$ $$\begin{array}{} \displaystyle u^m(t) = \sum_{k=1}^{m} \gamma_{k,m}(t) e_k\,, \; m = 1, 2, \ldots \end{array}$$

which is the approximate solutions of Faedo-Galerkin of order m, that is

$\begin{matrix} (\begin{matrix} 〈 \frac{d u^{m}}{d t}, e_{k} 〉 + 〈 Δ u^{m}, e_{k} 〉 = 〈 f (u^{m}), e_{k} 〉 + 〈 b (t, u_{t}^{m}), e_{k} 〉 + 〈 g, e_{k} 〉 \\ 〈 u^{m} (τ), e_{k} 〉 = 〈 P_{m} u^{0}, e_{k} 〉 = 〈 u^{0}, e_{k} 〉 i.e. P_{m} u^{m} (τ) \to u^{0} in L^{2} (Ω) \\ 〈 u^{m} (τ + θ), e_{k} 〉 = 〈 P_{m} φ (θ), e_{k} 〉 = 〈 φ (θ), e_{k} 〉 \forall θ \in (- r, 0) \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle \left\{ \begin{array}{l} \langle \frac{du^m}{dt},e_k\rangle + \langle \Delta u^m , e_k\rangle = \langle f(u^m), e_k\rangle+\langle b(t,u_{t}^{m}), e_k\rangle + \langle g,e_k\rangle \\ \langle u^m(\tau),e_k\rangle= \langle P_m u^0 ,e_k\rangle=\langle u^0,e_k\rangle \; \mbox{i.e.}\; P_mu^{m}(\tau) \to u^{0}\; \mbox{in}\; L^2(\Omega) \\ \langle u^{m}(\tau + \theta),e_k\rangle=\langle P_m \varphi(\theta),e_k\rangle=\langle \varphi(\theta),e_k\rangle \; \forall \theta \in (-r,0) \end{array}\right. \end{array}$$

for all k = 1 … m. Where γ_k,m(t) = 〈 u^m(t), e_k 〉 denote the Fourier coefficients; such that γ_m,k ∈ C¹((τ, T); ℝ) ∩ L²((τ–r, T), ℝ), $\begin{matrix} γ_{k, m}^{'} \end{matrix}$ $\begin{array}{} \gamma'_{k,m} \end{array}$ (t) is absolutely continuous, and P_mu(t) = $\begin{matrix} \sum_{k = 1}^{m} \end{matrix}$ $\begin{array}{} \sum^{m}_{k=1} \end{array}$ 〈u,e_k〉 e_k is the orthogonal projection of L²(Ω) and $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H^{1}_{0} \end{array}$ (Ω) in V_m = span{e₁, …, e_m}.

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.

claim 1

For all m ∈ ℕ^∗and all T > τ, the sequence {u^m} is bounded in

$\begin{matrix} L^{\infty} ((τ, T); L^{2} (Ω)) \cap L^{2} ((τ, T); H_{0}^{1} (Ω)) \cap L^{p} ((τ, T); L^{p} (Ω)) . \end{matrix}$ $$\begin{array}{} \displaystyle L^{\infty} ((\tau, T);L^2(\Omega))\cap L^{2} ((\tau, T);H^1_0(\Omega)) \cap L^{p} ((\tau, T);L^p(\Omega))\,. \end{array}$$

Multiplying (1.1) by u^m and integrating over Ω, we obtain

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ u^{m} (t) ∥^{2} + ∥ \nabla u^{m} (t) ∥^{2} = \int_{Ω} f (u^{m}) u^{m} + \int_{Ω} b (t, u_{t}^{m}) u^{m} + \int_{Ω} g u^{m} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2} \frac{d}{dt} \Vert u^m (t) \Vert^2 + \Vert \nabla u^m (t) \Vert^2 = \int_{\Omega} f(u^m)u^m + \int_{\Omega} b(t,u_t^m)u^m + \int_{\Omega} gu^m \,. \end{array}$$

Using the hypothesis (1.2) and the Young inequality, we get

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ u^{m} (t) ∥^{2} + ∥ \nabla u^{m} (t) ∥^{2} \\ \leq c | Ω | - μ_{1} ∥ u^{m} (t) ∥^{p} + \frac{1}{2} ∥ b (t, u_{t}^{m}) ∥^{2} + \frac{1}{2} ∥ u^{m} (t) ∥^{2} + \frac{1}{2} ∥ g (t) ∥^{2} + \frac{1}{2} ∥ u^{m} (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\frac{1}{2} \frac{d}{dt} \Vert u^m (t) \Vert^2 + \Vert \nabla u^m (t) \Vert^2 \\ \displaystyle \leq c\vert \Omega \vert - \mu_1\Vert u^m(t) \Vert^p + \frac{1}{2}\Vert b(t, u_t^m) \Vert^2 + \frac{1}{2} \Vert u^m(t) \Vert^2 + \frac{1}{2}\Vert g(t)\Vert^2 + \frac{1}{2}\Vert u^m(t) \Vert^2\,. \end{array}$$

So, one has

$\begin{matrix} \frac{d}{d t} ∥ u^{m} (t) ∥^{2} + 2 ∥ \nabla u^{m} (t) ∥^{2} + 2 μ_{1} ∥ u^{m} (t) ∥^{p} \\ \leq 2 c | Ω | + ∥ b (t, u_{t}^{m}) ∥^{2} + ∥ g (t) ∥^{2} + ∥ u^{m} (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\frac{d}{dt} \Vert u^m (t) \Vert^2 + 2\Vert \nabla u^m (t) \Vert^2 + 2\mu_1\Vert u^m(t) \Vert^p\\ \leq 2c\vert \Omega \vert + \Vert b(t, u_t^m) \Vert^2 + \Vert g(t)\Vert^2 + \Vert u^m(t) \Vert^2\,. \end{array}$$

After integrating this last estimate over [τ,t], τ ≤ t ≤ T, we use (II) and (IV), so we get

$\begin{matrix} ∥ u^{m} (t) ∥^{2} + 2 \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s + 2 μ_{1} \int_{τ}^{t} ∥ u^{m} (s) ∥^{p} d s \\ \leq 2 c | Ω | (t - τ) + ∥ u^{m} (τ) ∥^{2} + C_{b} \int_{τ - r}^{t} ∥ u^{m} (s) ∥^{2} d s \\ + \int_{τ}^{t} ∥ g (s) ∥^{2} d s + \int_{τ}^{t} ∥ u^{m} (s) ∥^{2} d s, \\ \leq 2 c | Ω | (t - τ) + ∥ u^{m} (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s + C_{b} \int_{τ}^{t} ∥ u^{m} (s) ∥^{2} d s \\ + \int_{τ}^{t} ∥ g (s) ∥^{2} d s + \int_{τ}^{t} ∥ u^{m} (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert u^m (t) \Vert^2 + 2\int_{\tau}^{t} \Vert \nabla u^m (s) \Vert^2 ds + 2\mu_1 \int_{\tau}^{t} \Vert u^m(s) \Vert^p ds \\ \displaystyle \leq 2c \vert \Omega \vert (t-\tau) + \Vert u^m (\tau) \Vert^2 + C_b \int_{\tau -r}^{t} \Vert u^m(s) \Vert^2 ds\\ \displaystyle + \int_{\tau}^{t} \Vert g(s)\Vert^2 ds + \int_{\tau}^{t} \Vert u^m(s) \Vert^2 ds \,,\\ \displaystyle\leq 2c\vert \Omega \vert (t-\tau) + \Vert u^m (\tau) \Vert^2 + C_b\int_{\tau -r}^{\tau} \Vert u^m(s) \Vert^2 ds + C_b\int_{\tau}^{t} \Vert u^m(s) \Vert^2 ds \\ \displaystyle + \int_{\tau}^{t} \Vert g(s)\Vert^2 ds + \int_{\tau}^{t} \Vert u^m(s) \Vert^2 ds \,.\end{array}$$

By the fact that λ₁ ‖ u‖ ² ≤ ‖ ∇ u ‖ ², one has

$\begin{matrix} ∥ u^{m} (t) ∥^{2} + 2 \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s + 2 μ_{1} \int_{τ}^{t} ∥ u^{m} (s) ∥^{p} d s \\ \leq 2 c | Ω | (t - τ) + ∥ u^{m} (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s + C_{b} λ_{1}^{- 1} \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s \\ + \int_{τ}^{t} ∥ g (s) ∥^{2} d s + λ_{1}^{- 1} \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert u^m (t) \Vert^2 + 2\int_{\tau}^{t} \Vert \nabla u^m (s) \Vert^2 ds + 2\mu_1\int_{\tau}^{t} \Vert u^m(s) \Vert^p ds\\ \displaystyle \leq 2c\vert \Omega \vert (t-\tau) + \Vert u^m (\tau) \Vert^2 + C_b\int_{\tau -r}^{\tau} \Vert u^m(s) \Vert^2 ds + C_b\lambda_1^{-1}\int_{\tau}^{t} \Vert \nabla u^m(s) \Vert^2 ds \\ \displaystyle + \int_{\tau}^{t} \Vert g(s)\Vert^2 ds + \lambda_1^{-1} \int_{\tau}^{t} \Vert \nabla u^m(s) \Vert^2 ds\,. \end{array}$$

Then, we find

$\begin{matrix} ∥ u^{m} (t) ∥^{2} + (2 - C_{b} λ_{1}^{- 1} - λ_{1}^{- 1}) \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s + 2 μ_{1} \int_{τ}^{t} ∥ u^{m} (s) ∥^{p} d s \\ \leq 2 c | Ω | (t - τ) + ∥ u^{m} (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s + \int_{τ}^{t} ∥ g (s) ∥^{2} d s, \\ \leq 2 c | Ω | (T - τ) + ∥ u^{m} (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s + \int_{τ}^{t} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert u^m (t) \Vert^2 + (2-C_b\lambda_1^{-1}-\lambda_1^{-1})\int_{\tau}^{t} \Vert \nabla u^m (s) \Vert^2 ds + 2\mu_1\int_{\tau}^{t} \Vert u^m(s) \Vert^p ds \\ \displaystyle \leq 2c\vert \Omega \vert (t-\tau) +\Vert u^m (\tau) \Vert^2 + C_b\int_{\tau -r}^{\tau} \Vert u^m(s) \Vert^2 ds + \int_{\tau}^{t} \Vert g(s)\Vert^2 ds \,,\\ \displaystyle \leq 2c\vert \Omega \vert (T-\tau) +\Vert u^m (\tau) \Vert^2 + C_b\int_{\tau -r}^{\tau} \Vert u^m(s) \Vert^2 ds + \int_{\tau}^{t} \Vert g(s)\Vert^2 ds \,. \end{array}$$(2.1)

Since g ∈ $\begin{matrix} L_{l o g}^{2} \end{matrix}$ $\begin{array}{} L^{2}_{log} \end{array}$ (ℝ,L²(Ω)) and for λ₁ > 1+ C_b/2, we deduce by this last estimate that for all T > τ, the sequence

$\begin{matrix} {u^{m}} is bounded in L^{\infty} ((τ, T); L^{2} (Ω)) \cap L^{2} ((τ, T); H_{0}^{1} (Ω)) \cap L^{p} ((τ, T); L^{p} (Ω)) . \end{matrix}$ $$\begin{array}{} \displaystyle \{u^m\}\;\mbox{ is bounded in}\; L^{\infty} ((\tau, T);L^2(\Omega))\cap L^{2} ((\tau, T);H^1_0(\Omega)) \cap L^{p} ((\tau, T);L^p(\Omega))\,. \end{array}$$(2.2)

Also, the estimate (2.1) implies that the local solution can extended to the interval [τ, T].

claim 2

$\begin{matrix} {f (u^{m})} i s b o u n d e d i n L^{q} ((τ, T); L^{q} (Ω)) . \end{matrix}$ $$\begin{array}{} \displaystyle \{f(u^m)\} \; {is\, bounded \,in}\; L^{q}((\tau, T);L^{q}(\Omega))\,. \end{array}$$(2.3)

Using (1.4), we have

$\begin{matrix} ∥ f (u^{m} (t) ∥_{L^{q} (Ω)}^{q} = \int_{Ω} | f (u^{m} (t, x)) |^{q} d x, \\ \leq l^{q} \int_{Ω} {(| u^{m} (t, x) |^{p - 1} + 1)}^{q} d x . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert f(u^m(t) \Vert_{L^q(\Omega)}^q = \int_{\Omega}\vert f(u^m(t,x)) \vert^q dx\,,\\ \displaystyle \qquad \qquad\qquad~ \leq l^q\int_{\Omega}\left(\vert u^m(t,x) \vert^{p-1} + 1\right)^q dx\,. \end{array}$$

By the convexity of the power and the fact that p = q(p–1), one has

$\begin{matrix} ∥ f (u^{m} (t)) ∥_{L^{q} (Ω)}^{q} \leq 2^{q - 1} l^{q} \int_{Ω} | u^{m} (t, x) |^{q (p - 1)} d x + 2^{q - 1} l^{q} | Ω |, \\ \leq 2^{q - 1} l^{q} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{q (p - 1)} + 2^{q - 1} l^{q} | Ω |, \\ \leq 2^{q - 1} l^{q} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{p} + 2^{q - 1} l^{q} | Ω | . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert f(u^m(t)) \Vert_{L^q(\Omega)}^q \leq 2^{q-1}l^q\int_{\Omega}\vert u^m(t,x) \vert^{q(p-1)} dx + 2^{q-1}l^q \vert \Omega \vert \,,\\ \displaystyle \qquad\qquad\qquad\quad\leq 2^{q-1}l^q \Vert u^m(t) \Vert_{L^p(\Omega)}^{q(p-1)} + 2^{q-1}l^q \vert \Omega \vert \,,\\ \displaystyle \qquad\qquad\qquad\quad\leq 2^{q-1}l^q \Vert u^m(t) \Vert_{L^p(\Omega)}^p + 2^{q-1}l^q \vert \Omega \vert\,. \end{array}$$

Integrating this last estimate over [τ,t], τ ≤ t ≤ T, one obtains

$\begin{matrix} \int_{τ}^{t} ∥ f (u^{m} (s)) ∥_{L^{q} (Ω)}^{q} d s \leq 2^{q - 1} l^{q} \int_{τ}^{t} ∥ u^{m} (s) ∥_{L^{p} (Ω)}^{p} d s + 2^{q - 1} l^{q} | Ω | (t - τ) \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau}^{t}\Vert f(u^m(s)) \Vert_{L^q(\Omega)}^q ds \leq 2^{q-1}l^q \int_{\tau}^{t} \Vert u^m(s) \Vert_{L^p(\Omega)}^p ds + 2^{q-1}l^q \vert \Omega \vert(t-\tau) \end{array}$$

From (2.1) we deduce that the term $\begin{matrix} \int_{τ}^{t} ∥ u^{m} (s) ∥_{L^{p} (Ω)}^{p} \end{matrix}$ $\begin{array}{} \int_{\tau}^{t} \Vert u^m(s) \Vert_{L^p(\Omega)}^p \end{array}$ds is bounded, so by this last estimate we conclude that {f(u^m)} is bounded in L^q((τ, T);L^q(Ω)), for all T > τ.

claim 3

$\begin{matrix} (\frac{\partial}{\partial t} u^{m}) \end{matrix}$ $\begin{array}{} \left\{\frac{\partial }{\partial t}u^m\right\} \end{array}$is bounded in L²((τ,T);L²(Ω)).

Now, multiplying (1.1) by $\begin{matrix} \frac{\partial u^{m}}{\partial t} \end{matrix}$ $\begin{array}{} \frac{\partial u^m}{\partial t} \end{array}$ and integrating over Ω, one has

$\begin{matrix} - {(\frac{d}{d t} u^{m} (t))}^{2} + \frac{1}{2} \frac{d}{d t} ∥ \nabla u^{m} (t) ∥^{2} \\ = \int_{Ω} f (u^{m}) \frac{\partial u^{m}}{\partial t} + \int_{Ω} b (t, u_{t}^{m}) \frac{\partial u^{m}}{\partial t} + \int_{Ω} g \frac{\partial u^{m}}{\partial t} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad-\left\Vert\frac{d}{d t} u^m(t) \right \Vert^2 + \frac{1}{2} \frac{d}{dt} \Vert \nabla u^m(t) \Vert^2 \nonumber\\ \displaystyle = \int_{\Omega} f(u^m)\frac{\partial u^m}{\partial t}+ \int_{\Omega} b(t,u^m_t)\frac{\partial u^m}{\partial t} + \int_{\Omega} g\frac{\partial u^m}{\partial t}\,. \end{array}$$(2.4)

On the other hand, we have

$\begin{matrix} \frac{d}{d t} F (u) = \frac{d F}{d u} \frac{\partial u}{\partial t}, \\ = f (u) \frac{\partial u}{\partial t} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{d t} F(u) = \frac{dF}{d u} \,\frac{\partial u}{\partial t}\,,\\ \displaystyle \qquad\qquad= f(u) \, \frac{\partial u}{\partial t} \,. \end{array}$$(2.5)

$\begin{matrix} \frac{d}{d t} \int_{Ω} F (u) = \int_{Ω} f (u) \frac{\partial u}{\partial t} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{d t}\int_{\Omega} F(u) = \int_{\Omega} f(u)\frac{\partial u}{\partial t} \,. \end{array}$$

Using this last equality in (2.4), we find

$\begin{matrix} {(\frac{d}{d t} u^{m} (t))}^{2} + \frac{1}{2} \frac{d}{d t} ∥ \nabla u^{m} (t) ∥^{2} = \frac{d}{d t} \int_{Ω} F (u^{m}) + \int_{Ω} b (t, u_{t}^{m}) \frac{\partial u^{m}}{\partial t} + \int_{Ω} g \frac{\partial u^{m}}{\partial t} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert\frac{d}{d t} u^m(t) \right \Vert^2 + \frac{1}{2} \frac{d}{dt} \Vert \nabla u^m(t) \Vert^2 = \frac{d}{dt} \int_{\Omega} F(u^m)+ \int_{\Omega} b(t,u^m_t)\frac{\partial u^m}{\partial t} + \int_{\Omega} g\frac{\partial u^m}{\partial t}\,. \end{array}$$

From (1.5) and Cauchy inequality, we have

$\begin{matrix} {(\frac{d}{d t} u^{m} (t))}^{2} + \frac{1}{2} \frac{d}{d t} ∥ \nabla u^{m} (t) ∥^{2}, \\ \leq \frac{d}{d t} \int_{Ω} (c^{'} - μ_{1}^{'} | u (t, x) |^{p}) d x + \frac{ε_{1}}{2} ∥ b (t, u_{t}^{m}) ∥^{2} + \frac{1}{2 ε_{1}} {(\frac{d}{d t} u^{m} (t))}^{2} \\ + \frac{ε_{2}}{2} ∥ g (t) ∥^{2} + \frac{1}{2 ε_{2}} {(\frac{d}{d t} u^{m} (t))}^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\left\Vert\frac{d}{d t} u^m(t) \right \Vert^2 + \frac{1}{2} \frac{d}{dt} \Vert \nabla u^m(t) \Vert^2 \,,\\ \displaystyle \leq \frac{d}{dt} \int_{\Omega} (c' - \mu'_1 \vert u(t,x) \vert^p) dx + \frac{\varepsilon_1}{2}\Vert b(t,u^m_t) \Vert^2 + \frac{1}{2 \varepsilon_1}\left\Vert \frac{d }{d t}u^m(t)\right\Vert^2 \\ \displaystyle + \frac{\varepsilon_2}{2}\Vert g(t) \Vert^2 + \frac{1}{2 \varepsilon_2}\left \Vert \frac{d}{d t} u^m(t) \right \Vert^2 \,. \end{array}$$

After simplification, one obtains

$\begin{matrix} (2 - \frac{1}{ε_{1}} - \frac{1}{ε_{2}}) {(\frac{d}{d t} u^{m} (t))}^{2} + \frac{d}{d t} (∥ \nabla u^{m} (t) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{p}) \\ \leq ε_{1} ∥ b (t, u_{t}^{m}) ∥^{2} + ε_{2} ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\left (2-\frac{1}{ \varepsilon_1}-\frac{1}{ \varepsilon_2}\right)\left\Vert\frac{d}{d t} u^m(t) \right\Vert^2 + \frac{d}{dt} \left(\Vert \nabla u^m(t) \Vert^2 + 2 \mu'_1 \Vert u^m(t) \Vert^p_{L^p(\Omega)}\right) \\ \leq {\varepsilon_1}\Vert b(t,u^m_t) \Vert^2 + {\varepsilon_2}\Vert g(t) \Vert^2 \,. \end{array}$$

We can choose ε₁ = ε₂ = 2 to get

$\begin{matrix} {(\frac{d}{d t} u^{m} (t))}^{2} + \frac{d}{d t} (∥ \nabla u^{m} (t) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{p}) \leq 2 ∥ b (t, u_{t}^{m}) ∥^{2} + 2 ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \left\Vert\frac{d}{d t} u^m(t) \right\Vert^2 + \frac{d}{dt} \left(\Vert \nabla u^m(t) \Vert^2 + 2 \mu'_1 \Vert u^m(t) \Vert^p_{L^p(\Omega)}\right) \leq 2\Vert b(t,u^m_t) \Vert^2 + 2\Vert g(t) \Vert^2 \,. \end{array} $$

Integrating this last estimate over [τ,t] and using (II) and (IV), one has

$\begin{matrix} \int_{τ}^{t} {(\frac{d}{d s} u^{m} (s))}^{2} d s + ∥ \nabla u^{m} (t) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{p} \\ \leq ∥ \nabla u^{m} (τ) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (τ) ∥_{L^{p} (Ω)}^{p} + 2 C_{b} \int_{τ - r}^{t} ∥ u^{m} (s) ∥^{2} d s + 2 \int_{τ}^{t} ∥ g (s) ∥^{2} d s, \\ \leq ∥ \nabla u^{m} (τ) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (τ) ∥_{L^{p} (Ω)}^{p} + 2 C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s \\ + 2 C_{b} \int_{τ}^{t} ∥ u^{m} (s) ∥^{2} d s + 2 \int_{τ}^{t} ∥ g (s) ∥^{2} d s \end{matrix}$ $$\begin{array}{} \displaystyle \quad \int_{\tau}^{t}\left\Vert\frac{d}{d s} u^m(s) \right\Vert^2 ds +\Vert \nabla u^m(t) \Vert^2 + 2 \mu'_1 \Vert u^m(t) \Vert^p_{L^p(\Omega)} \\ \displaystyle \leq \Vert \nabla u^m(\tau) \Vert^2 + 2 \mu'_1 \Vert u^m(\tau) \Vert^p_{L^p(\Omega)} + 2C_b\int_{\tau -r}^{t}\Vert u^m(s) \Vert^2 ds+ 2 \int_{\tau}^{t} \Vert g(s) \Vert^2 ds \,,\\ \displaystyle \leq \Vert \nabla u^m(\tau) \Vert^2 + 2 \mu'_1 \Vert u^m(\tau) \Vert^p_{L^p(\Omega)} + 2C_b\int_{\tau -r}^{\tau}\Vert u^m(s) \Vert^2 ds \\ \displaystyle + 2C_b\int_{\tau}^{t}\Vert u^m(s) \Vert^2 ds+ 2 \int_{\tau}^{t} \Vert g(s) \Vert^2 ds \end{array}$$

Since λ₁ ‖ u ‖² ≤ ‖ ∇u‖², one has

$\begin{matrix} \int_{τ}^{t} {(\frac{d}{d s} u^{m} (s))}^{2} d s + ∥ \nabla u^{m} (t) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (t) ∥_{L^{p} (Ω)}^{p} \\ \leq ∥ \nabla u^{m} (τ) ∥^{2} + 2 μ_{1}^{'} ∥ u^{m} (τ) ∥_{L^{p} (Ω)}^{p} + 2 C_{b} \int_{τ - r}^{τ} ∥ u^{m} (s) ∥^{2} d s \\ + 2 C_{b} λ_{1}^{- 1} \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s + 2 \int_{τ}^{t} ∥ g (s) ∥^{2} d s \end{matrix}$ $$\begin{array}{} \displaystyle \quad\int_{\tau}^{t}\left\Vert\frac{d}{d s} u^m(s) \right\Vert^2 ds +\Vert \nabla u^m(t) \Vert^2 + 2 \mu'_1 \Vert u^m(t) \Vert^p_{L^p(\Omega)} \\ \displaystyle \leq \Vert \nabla u^m(\tau) \Vert^2 + 2 \mu'_1 \Vert u^m(\tau) \Vert^p_{L^p(\Omega)} + 2C_b\int_{\tau -r}^{\tau}\Vert u^m(s) \Vert^2 ds \\ \displaystyle + 2C_b \lambda_1^{-1}\int_{\tau}^{t}\Vert \nabla u^m(s) \Vert^2 ds + 2 \int_{\tau}^{t} \Vert g(s) \Vert^2 ds \end{array}$$

From (2.1), we have $\begin{matrix} \int_{τ}^{t} ∥ \nabla u^{m} (s) ∥^{2} d s \end{matrix}$ $\begin{array}{} \int_{\tau}^{t}\Vert \nabla u^m(s) \Vert^2 ds \end{array}$

is bounded and since g ∈ $\begin{matrix} L_{l o c}^{2} \end{matrix}$ $\begin{array}{} L^2_{loc} \end{array}$ (ℝ;L²(Ω)), this last estimate gives that

$\begin{matrix} (\frac{\partial}{\partial t} u^{m}) is bounded in L^{2} ((τ, T); L^{2} (Ω)), \end{matrix}$ $$\begin{array}{} \displaystyle \left\{\frac{\partial }{\partial t}u^m\right\} \; \mbox{is bounded in}\; L^2((\tau,T);L^2(\Omega)) \,, \end{array}$$

for all T > τ.

From the claims (1), (2) and (3), the hypothesis (IV) and the remark (1)1, we can extract a subsequence (relabelled the same) such that

$\begin{matrix} u^{m} ⇀ u weakly* in L^{\infty} ((τ, T); L^{2} (Ω)), \\ u^{m} ⇀ u weakly in L^{2} ((τ, T); H_{0}^{1} (Ω)), \\ u^{m} ⇀ u weakly in L^{p} ((τ, T); L^{p} (Ω)), \\ \frac{\partial u^{m}}{\partial t} \to \frac{\partial u}{\partial t} strongly in L^{2} ((τ, T); L^{2} (Ω)), \\ f (u^{m}) ⇀ σ^{'} weakly in L^{q} ((τ, T); L^{q} (Ω)), \\ b (., u_{.}^{m}) \to b (., u_{.}) strongly in L^{2} ((τ, T); L^{2} (Ω)) . \end{matrix}$ $$\begin{array}{} \displaystyle u^m \rightharpoonup u \;\mbox{weakly* in}\; L^{\infty}((\tau,T);L^2(\Omega)), \\ \displaystyle u^m \rightharpoonup u \;\mbox{weakly in} \; L^{2}((\tau,T);H^{1}_0(\Omega)),\\ \displaystyle u^m \rightharpoonup u \;\mbox{weakly in} \; L^{p}((\tau,T);L^p(\Omega)),\\ \displaystyle \frac{\partial u^m}{\partial t} \rightarrow \frac{\partial u}{\partial t} \;\mbox{strongly in}\; L^{2}((\tau,T);L^2(\Omega)),\\ \displaystyle f(u^m) \rightharpoonup \sigma' \;\mbox{weakly in}\; L^{q}((\tau,T);L^q(\Omega)),\\ \displaystyle b(.,u^m_.) \rightarrow b(.,u_.) \;\mbox{strongly in}\; L^{2}((\tau,T);L^2(\Omega))\,. \end{array}$$(2.6)

By the Aubin-Lions lemma of compactness, we conclude that u^m → u strongly in L²((τ,T);L²(Ω)). Thus u^m → u a.e [τ,T]× Ω.

Since f is continuous, we deduce that f(u^m) → 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); $\begin{matrix} H_{0}^{1} \end{matrix}$ $\begin{array}{} \displaystyle H^{1}_{0} \end{array}$ (Ω)) such that v(T) = 0 and we note from (1.1) that

$\begin{matrix} \int_{τ}^{T} - 〈 u, v^{'} 〉 + \int_{τ}^{T} \int_{Ω} \nabla u \nabla v = \int_{τ}^{T} \int_{Ω} f (u) v + \int_{τ}^{T} 〈 b (t, u_{t}), v 〉 \\ + \int_{τ}^{T} \int_{Ω} g v + 〈 u (τ), v (τ) 〉 . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{T}_{\tau} - \langle u,v'\rangle + \int_{\tau}^{T}\int_{\Omega} \nabla u \nabla v = \int_{\tau}^{T}\int_{\Omega} f(u)v + \int_{\tau}^{T} \langle b(t,u_t), v\rangle \\ \displaystyle \qquad \qquad\qquad\qquad\qquad\qquad~+ \int_{\tau}^{T}\int_{\Omega} gv + \langle u(\tau),v(\tau)\rangle \,. \end{array}$$(2.7)

In a similar way, from the Faedo-Galerkin approximations, we have

$\begin{matrix} \int_{τ}^{T} - 〈 u^{m}, v^{'} 〉 \int_{τ}^{T} \int_{Ω} \nabla u^{m} \nabla v = \int_{τ}^{T} \int_{Ω} f (u^{m}) v + \int_{τ}^{T} 〈 b (t, u_{t}^{m}), v 〉 \\ + \int_{τ}^{T} \int_{Ω} g v + 〈 u^{m} (τ), v (τ) 〉 . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{T}_{\tau} - \langle u^m,v'\rangle \int_{\tau}^{T}\int_{\Omega} \nabla u^m \nabla v = \int_{\tau}^{T}\int_{\Omega} f(u^m)v + \int_{\tau}^{T} \langle b(t,u^m_t), v\rangle \\ \displaystyle \qquad\qquad\qquad\qquad\qquad\qquad\quad + \int_{\tau}^{T}\int_{\Omega} gv + \langle u^m(\tau),v(\tau)\rangle \,. \end{array}$$(2.8)

Using the fact that u^m(τ) → u⁰ in L²(Ω) and (2.6) to find

$\begin{matrix} \int_{τ}^{T} - 〈 u, v^{'} 〉 + \int_{τ}^{T} \int_{Ω} \nabla u \nabla v = \int_{τ}^{T} \int_{Ω} f (u) v + \int_{τ}^{T} 〈 b (t, u_{t}), v 〉 \\ + \int_{τ}^{T} \int_{Ω} g v + 〈 u^{0}, v (τ) 〉 . \end{matrix}$ $$\begin{array}{} \displaystyle \int^{T}_{\tau} - \langle u,v'\rangle + \int_{\tau}^{T}\int_{\Omega} \nabla u \nabla v = \int_{\tau}^{T}\int_{\Omega} f(u)v + \int_{\tau}^{T} \langle b(t,u_t), v\rangle \\ \displaystyle \qquad\qquad\qquad\qquad\qquad\qquad~+ \int_{\tau}^{T}\int_{\Omega} gv + \langle u^0,v(\tau)\rangle \,. \end{array}$$(2.9)

Since v(τ) is given arbitrarily, comparing (2.7) and (2.9) we deduce that u(τ) = u⁰.

To prove that u ∈ C([τ,T];L²(Ω)), we put w^m = u^m–u then we have

$\begin{matrix} \frac{\partial}{\partial t} w^{m} - Δ w^{m} = f (u^{m}) - f (u) + b (t, u_{t}^{m}) - b (t, u_{t}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial}{\partial t} w^m - \Delta w^m = f(u^m)-f(u) + b(t,u^m_t) - b(t,u_t) \,. \end{array}$$

Multiplying this equation by w^m and integrating over Ω, we obtain

$\begin{matrix} \frac{d}{d t} ∥ w^{m} (t) ∥^{2} + 2 ∥ \nabla w^{m} (t) ∥^{2} = 2 \int_{Ω} (f (u^{m}) - f (u)) w^{m} \\ + 2 \int_{Ω} (b (t, u_{t}^{m}) - b (t, u_{t})) (u^{m} - u) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt} \Vert w^m (t) \Vert^2 + 2\Vert \nabla w^m (t) \Vert^2 = 2 \int_{\Omega}\left(f(u^m)-f(u)\right)w^m \\ \displaystyle \qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,\,\,+ 2\int_{\Omega}(b(t,u_t^m)-b(t,u_t))(u^m-u) \,. \end{array}$$

By (1.3), (I) and (1.6), we get

$\begin{matrix} \frac{d}{d t} ∥ w^{m} (t) ∥^{2} + 2 ∥ \nabla w^{m} (t) ∥^{2} \leq 2 k ∥ w^{m} (t) ∥^{2} + 2 L_{b} ∥ w_{t}^{m} ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt} \Vert w^m (t) \Vert^2 + 2\Vert \nabla w^m (t) \Vert^2 \leq 2 k \Vert w^m(t) \Vert^2 + 2 L_b \Vert w^m_t \Vert^2_{L^2([-r,0];L^2(\Omega))} \,. \end{array}$$

Integrating over [τ,t], we get

$\begin{matrix} ∥ w^{m} (t) ∥^{2} - ∥ w^{m} (τ) ∥^{2} + 2 \int_{τ}^{t} ∥ \nabla w^{m} (s) ∥^{2} d s \\ \leq 2 k \int_{τ}^{t} ∥ w^{m} (s) ∥^{2} + 2 L_{b} \int_{τ}^{t} \int_{- r}^{0} ∥ w^{m} (s + θ) ∥^{2} d θ d s, \\ \leq 2 k \int_{τ}^{t} ∥ w^{m} (s) ∥^{2} + 2 L_{b} \int_{- r}^{0} \int_{τ - r}^{t} ∥ w^{m} (s) ∥^{2} d s d θ, \\ \leq 2 k \int_{τ}^{t} ∥ w^{m} (s) ∥^{2} + 2 L_{b} r \int_{τ - r}^{τ} ∥ w^{m} (s) ∥^{2} d s + 2 L_{b} r \int_{τ}^{t} ∥ w^{m} (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert w^m (t) \Vert^2 - \Vert w^m (\tau) \Vert^2 + 2\int_{\tau}^{t} \Vert \nabla w^m (s) \Vert^2 ds\\ \displaystyle \leq 2k \int_{\tau}^{t}\Vert w^m(s) \Vert^2 + 2 L_b \int_{\tau}^{t}\int_{-r}^{0}\Vert w^m(s+\theta) \Vert^2 d\,\theta ds \,,\\ \displaystyle \leq 2k \int_{\tau}^{t}\Vert w^m(s) \Vert^2 + 2 L_b \int_{-r}^{0} \int_{\tau-r}^{t}\Vert w^m(s) \Vert^2 ds d\theta \,,\\ \displaystyle \leq 2k \int_{\tau}^{t}\Vert w^m(s) \Vert^2 + 2 L_br \int_{\tau-r}^{\tau}\Vert w^m(s) \Vert^2 ds + 2 L_br \int_{\tau}^{t}\Vert w^m(s) \Vert^2 ds\,. \end{array} $$

Therefore by by this last estimate, we can deduce that

$\begin{matrix} ∥ w^{m} (t) ∥^{2} \leq ∥ w^{m} (τ) ∥^{2} + 2 L_{b} r \int_{τ - r}^{τ} ∥ w^{m} (s) ∥^{2} d s + (2 k + 2 L_{b} r) \int_{τ}^{t} ∥ w^{m} (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w^m (t) \Vert^2 \leq \Vert w^m (\tau) \Vert^2 + 2 L_br \int_{\tau-r}^{\tau}\Vert w^m(s) \Vert^2 ds + (2k+2 L_br) \int_{\tau}^{t}\Vert w^m(s) \Vert^2 ds\,. \end{array}$$

Applying the Gronwall lemma to this estimate, we obtain

$\begin{matrix} ∥ w^{m} (t) ∥^{2} \leq (∥ w^{m} (τ) ∥^{2} + 2 L_{b} r \int_{τ - r}^{τ} ∥ w^{m} (s) ∥^{2} d s) e^{(2 k + 2 L_{b} r) (t - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w^m (t) \Vert^2 \leq \left(\Vert w^m (\tau) \Vert^2 + 2 L_br \int_{\tau-r}^{\tau}\Vert w^m(s) \Vert^2 ds\right)e^{(2k+2 L_br) (t-\tau)}\,. \end{array}$$(2.10)

Since u^m(τ)→ u⁰ and u^m(τ + θ)→ φ(θ), the estimate (2.10) shows that u^m → 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 u^0,1, u^0,2 and φ¹, φ². Denoting that w = u¹ – u² and repeating the argument as in the proof of (2.10), we find

$\begin{matrix} ∥ w (t) ∥^{2} \leq (∥ w (τ) ∥^{2} + 2 L_{b} r \int_{τ - r}^{τ} ∥ w (s) ∥^{2} d s) e^{(2 k + 2 L_{b} r) (t - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w (t) \Vert^2 \leq \left(\Vert w (\tau) \Vert^2 + 2 L_br \int_{\tau-r}^{\tau}\Vert w(s) \Vert^2 ds\right)e^{(2k+2 L_br) (t-\tau)}\,. \end{array}$$(2.11)

and this completes the proof of the theorem. ◼

Existence of pullback D-attractors

3.1

Preliminaries of pullback D-attractors

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

$\begin{matrix} S (t, τ) D (τ) \subset B (t) f o r a l l τ \leq τ_{0} (t, \hat{D}) . \end{matrix}$ $$\begin{array}{} \displaystyle S(t,\tau)D(\tau) \subset B(t)\quad \it{for\, all }\; \tau \leq \tau_0(t,\widehat{D}) \,. \end{array}$$

Definition 4

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 x_n ∈ D(τ_n), the sequence {S(t,τ_n)x_n} is relatively compact in X.

Definition 5

A family Â = {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;

Â is invariant; i.e., S(t,τ)A(τ) = A(t), for all t ≥ τ;

Â is pullback 𝒟-attracting; i.e.,

$\begin{matrix} lim_{τ \to - \infty} d i s t (S (t, τ) D (τ), A (t)) = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \lim_{\tau \to -\infty} dist(S(t,\tau)D(\tau), A(t))=0\,, \end{array}$$

for all D̂ ∈ 𝒟 and all t ∈ ℝ;

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

$\begin{matrix} A (t) = ⋂_{s \leq t} \bar{⋃_{τ \leq s} S (t, τ) B (τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle A(t) = \bigcap_{s\leq t}\overline{\bigcup_{\tau \leq s}S(t,\tau) B(\tau)}\,. \end{array}$$

3.2

Construction of the associated process

Now, we will apply the above results in the phase space H: = L²(Ω) × L²([–r,0]; L²(Ω)), which is a Hilbert space with the norm

$\begin{matrix} ∥ (u^{0}, φ) ∥_{H}^{2} = ∥ \nabla u^{0} ∥^{2} + \int_{- r}^{0} ∥ φ (θ) ∥^{2} d θ, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert (u^0,\varphi) \Vert^2_H = \Vert \nabla u^0 \Vert^2 + \int_{-r}^{0}\Vert \varphi (\theta) \Vert^2 d\theta \,, \end{array}$$

with (u⁰,φ)∈ H. To this aim, We consider g ∈ $\begin{matrix} L_{l o c}^{2} \end{matrix}$ $\begin{array}{} L^{2}_{loc} \end{array}$ (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

$\begin{matrix} S (t, τ) : H \to H \\ (u^{0}, φ) ⟼ S (t, τ) (u^{0}, φ) = (u (t), u_{t}), \end{matrix}$ $$\begin{array}{} \displaystyle S(t,\tau) : H\rightarrow H \\ \displaystyle \qquad\quad~~(u^0,\varphi) \longmapsto S(t,\tau)(u^0,\varphi) = (u(t),u_t) \,, \end{array} $$(3.1)

with t ≥ τ, τ ∈ ℝ and u is the weak solution to (1.1), defines a process.

On the other hand, we construct the family of mappings

$\begin{matrix} U (t, τ) : H \to C ([- r, 0]; L^{2} (Ω)) \\ (u^{0}, φ) ⟼ U (t, τ) (u^{0}, φ) = u_{t}, \forall t \geq τ + r, \end{matrix}$ $$\begin{array}{} \displaystyle U(t,\tau) : H\rightarrow C([-r,0];L^2(\Omega)) \\ \displaystyle \qquad\qquad (u^0,\varphi) \longmapsto U(t,\tau)(u^0,\varphi) = u_t \,,\; \forall t \geq \tau + r\,, \end{array}$$(3.2)

which we will use in our analysis. Of course, it is sensible to expect that the both operators should be related. Let us consider the linear mapping

$\begin{matrix} j : C ([- r, 0]; L^{2} (Ω)) \to l^{2} (Ω) \times C ([- r, 0]; L^{2} (Ω)) \\ φ ⟼ j (φ) = (φ (0), φ) . \end{matrix}$ $$\begin{array}{} \displaystyle j : C([-r,0]; L^2(\Omega))\rightarrow l^2(\Omega) \times C([-r,0]; L^2(\Omega)) \\ \displaystyle ~~~~~\varphi \longmapsto j(\varphi) = (\varphi(0),\varphi) \,. \end{array}$$

This map is obviously continuous from C([–r,0]; L²(Ω)) into H. We note that for all (u⁰,φ) ∈ H provided that t ≥ τ + r, so we write

$\begin{matrix} S (t, τ) (u^{0}, φ) = j (U (t, τ) (u^{0}, φ)), \forall (u^{0}, φ) \in H, \forall t \geq τ + r . \end{matrix}$ $$\begin{array}{} \displaystyle S(t,\tau)(u^0,\varphi) = j(U(t,\tau)(u^0,\varphi))\,,\; \forall (u^0,\varphi) \in H\,,\; \forall t \geq \tau + r\,. \end{array}$$

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: = $\begin{matrix} 2 (\frac{1}{2} + k + \frac{C_{b}}{2} - λ_{1}) > 0, \end{matrix}$ $\begin{array}{} 2(\frac{1}{2}+k+\frac{C_b}{2}-\lambda_1)>0\,, \end{array}$such that

$\begin{matrix} ∥ u (t) - v (t) ∥^{2} \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥^{2}) e^{ν (t - τ)}, \forall t \geq τ . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u(t)-v(t) \Vert^2 \leq \left(\Vert u^0-v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert^2\right) e^{\nu (t-\tau)}\,,\; \forall t\geq \tau\,. \end{array}$$(3.3)

It also holds

$\begin{matrix} ∥ u_{t} - v_{t} ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥^{2}) e^{ν (t - r - τ)}, \forall t \geq τ + r . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_t-v_t \Vert_{C([-r,0];L^2(\Omega))}^2 \leq \left(\Vert u^0-v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert^2\right) e^{\nu (t-r-\tau)}\,,\; \forall t\geq \tau + r\,. \end{array}$$(3.4)

Proof

From (1.1), one has

$\begin{matrix} \frac{\partial}{\partial t} (u - v) - Δ (u - v) = f (u) - f (v) + b (t, u_{t}) - b (t, v_{t}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial}{\partial t} (u-v) - \Delta (u-v) = f(u) -f(v) + b(t,u_t) - b(t,v_t)\,. \end{array}$$

We put w = u–v, we obtain

$\begin{matrix} \frac{\partial w}{\partial t} - Δ w = f (u) - f (v) + b (t, u_{t}) - b (t, v_{t}) . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{\partial w}{\partial t} - \Delta w = f(u) -f(v) + b(t,u_t) - b(t,v_t)\,. \end{array}$$

Multiplying this equation by w and integrating it over Ω, one gets

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ w (t) ∥^{2} + ∥ \nabla w (t) ∥^{2} = \int_{Ω} (f (u) - f (v)) w + \int_{Ω} (b (t, u_{t}) - b (t, v_{t})) w . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2} \frac{d}{dt} \Vert w(t) \Vert^2 + \Vert \nabla w(t) \Vert^2 = \int_{\Omega} \left( f(u)-f(v)\right)w + \int_{\Omega} \left( b(t,u_t) - b(t,v_t)\right)w\,. \end{array}$$

Using (1.3) and Cauchy-Schwarz inequality, one has

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ w (t) ∥^{2} + ∥ \nabla w (t) ∥^{2} \leq k ∥ w (t) ∥^{2} + ∥ b (t, u_{t}) - b (t, v_{t}) ∥ ∥ w (t) ∥ . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2} \frac{d}{dt} \Vert w(t) \Vert^2 + \Vert \nabla w(t) \Vert^2 \leq k\Vert w(t) \Vert^2 + \Vert b(t,u_t) - b(t,v_t) \Vert \Vert w(t) \Vert\,. \end{array}$$

Since λ₁ ‖ w(t) ‖² ≤ ‖ ∇ w(t) ‖² and by the Young inequality, one finds

$\begin{matrix} \frac{d}{d t} ∥ w (t) ∥^{2} + 2 λ_{1} ∥ w (t) ∥^{2} \leq \frac{d}{d t} ∥ w (t) ∥^{2} + 2 ∥ \nabla w (t) ∥^{2}, \\ \leq 2 k ∥ w (t) ∥^{2} + ∥ b (t, u_{t}) - b (t, v_{t}) ∥^{2} + ∥ w (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt} \Vert w(t) \Vert^2 + 2 \lambda_1\Vert w(t) \Vert^2 \leq \frac{d}{dt} \Vert w(t) \Vert^2 + 2 \Vert \nabla w(t) \Vert^2\,,\\ \displaystyle \qquad\qquad\qquad\qquad\qquad\quad\leq 2k\Vert w(t) \Vert^2 + \Vert b(t,u_t) - b(t,v_t) \Vert^2 + \Vert w(t) \Vert^2 \,. \end{array}$$

Therefore, one has

$\begin{matrix} \frac{d}{d t} ∥ w (t) ∥^{2} \leq 2 (\frac{1}{2} + k - λ_{1}) ∥ w (t) ∥^{2} + ∥ b (t, u_{t}) - b (t, v_{t}) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt} \Vert w(t) \Vert^2 \leq 2\left(\frac{1}{2}+k -\lambda_1\right) \Vert w(t) \Vert^2 + \Vert b(t,u_t) - b(t,v_t) \Vert^2\,. \end{array}$$

Integrating this last estimate from τ to t and using (1.7), one obtains

$\begin{matrix} ∥ w (t) ∥^{2} \leq ∥ w (τ) ∥^{2} + 2 (\frac{1}{2} + k - λ_{1}) \int_{τ}^{t} ∥ w (s) ∥^{2} d s \\ + \int_{τ}^{t} ∥ b (s, u_{s}) - b (s, v_{s}) ∥^{2} d s, \\ \leq ∥ w (τ) ∥^{2} + 2 (\frac{1}{2} + k - λ_{1}) \int_{τ}^{t} ∥ w (s) ∥^{2} d s + C_{b} \int_{τ - r}^{t} ∥ w (s) ∥^{2} d s, \\ \leq ∥ w (τ) ∥^{2} + 2 (\frac{1}{2} + k - λ_{1}) \int_{τ}^{t} ∥ w (s) ∥^{2} d s + C_{b} \int_{τ - r}^{τ} ∥ w (s) ∥^{2} d s \\ + C_{b} \int_{τ}^{t} ∥ w (s) ∥^{2} d s, \\ \leq ∥ w (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ w (s) ∥^{2} d s + 2 (\frac{1}{2} + k + \frac{C_{b}}{2} - λ_{1}) \int_{τ}^{t} ∥ w (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w(t) \Vert^2 \leq \Vert w(\tau) \Vert^2 + 2 \left(\frac{1}{2}+k -\lambda_1\right) \int_{\tau}^{t} \Vert w(s) \Vert^2 ds \\ \displaystyle \qquad\quad~+ \int_{\tau}^{t} \Vert b(s,u_s) - b(s,v_s) \Vert^2 ds\,,\\ \displaystyle \qquad\quad~\leq \Vert w(\tau) \Vert^2 + 2 \left(\frac{1}{2}+k -\lambda_1\right) \int_{\tau}^{t} \Vert w(s) \Vert^2 ds + C_b\int_{\tau -r}^{t} \Vert w(s) \Vert^2 ds\,,\\ \displaystyle \qquad\quad~\leq \Vert w(\tau) \Vert^2 + 2 \left(\frac{1}{2}+k -\lambda_1\right) \int_{\tau}^{t} \Vert w(s) \Vert^2 ds + C_b\int_{\tau -r}^{\tau} \Vert w(s) \Vert^2 ds \\ \displaystyle \qquad\quad~+ C_b \int_{\tau }^{t} \Vert w(s) \Vert^2 ds\,,\\ \displaystyle \qquad\quad~\leq \Vert w(\tau) \Vert^2 + C_b\int_{\tau -r}^{\tau} \Vert w(s) \Vert^2 ds + 2 \left(\frac{1}{2}+k + \frac{C_b}{2}-\lambda_1\right) \int_{\tau}^{t} \Vert w(s) \Vert^2 ds\,. \end{array}$$

By the Gronwall lemma, for all t ≥ τ, one deduces

$\begin{matrix} ∥ w (t) ∥^{2} \leq (∥ w (τ) ∥^{2} + C_{b} \int_{τ - r}^{τ} ∥ w (s) ∥^{2} d s) e^{ν (t - τ)}, \\ \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2}) e^{ν (t - τ)}, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w(t) \Vert^2 \leq \left(\Vert w(\tau) \Vert^2 + C_b \int_{\tau -r}^{\tau} \Vert w(s) \Vert^2 ds\right) e^{\nu (t-\tau)}\,,\\ \displaystyle \qquad\quad~ \leq \left(\Vert u^0 -v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert_{L^2([-r,0];L^2(\Omega))}^2 \right) e^{\nu (t-\tau)}\,, \end{array}$$

and by this last estimate, we proved (3.3). Now, assume that t ≥ τ + r, so t + θ ≥ τ for all θ ∈ [–r,0] and one has

$\begin{matrix} ∥ w (t + θ) ∥^{2} \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2}) e^{ν (t + θ - τ)}, \\ \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2}) e^{ν (t - r - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w(t+\theta) \Vert^2 \leq \left(\Vert u^0 -v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert_{L^2([-r,0];L^2(\Omega))}^2 \right) e^{\nu (t+\theta-\tau)}\,,\\ \displaystyle \qquad\qquad~~~~\leq \left(\Vert u^0 -v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert_{L^2([-r,0];L^2(\Omega))}^2 \right) e^{\nu (t-r-\tau)}\,. \end{array}$$

Hence, we conclude

$\begin{matrix} ∥ w_{t} ∥_{C ([- r, 0]; L^{2} (Ω))} \leq (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2}) e^{ν (t - r - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert w_t \Vert_{C([-r,0];L^2(\Omega))} \leq \left(\Vert u^0 -v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert_{L^2([-r,0];L^2(\Omega))}^2 \right) e^{\nu (t-r-\tau)}\,. \end{array}$$

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)

$\begin{matrix} ∥ u_{t} - v_{t} ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} = \int_{- r}^{0} ∥ u (t + θ) - v (t + θ) ∥^{2} d θ, \\ \leq \int_{- r}^{0} sup_{s \in [- r, 0]} ∥ u (t + s) - v (t + s) ∥^{2} d θ, \\ \leq r (∥ u^{0} - v^{0} ∥^{2} + C_{b} ∥ φ - ϕ ∥^{2}) e^{ν (t - r - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_t -v_t \Vert^2_{L^2([-r,0];L^2(\Omega))} = \int_{-r}^{0} \Vert u(t+\theta) - v(t+\theta) \Vert^2 d\theta\,,\\ \displaystyle \qquad\qquad\qquad\qquad\quad~\leq \int_{-r}^{0} \sup_{s\in [-r,0]}\Vert u(t+s) - v(t+s) \Vert^2 d\theta\,,\\ \displaystyle \qquad\qquad\qquad\qquad\quad~\leq r \left(\Vert u^0-v^0 \Vert^2 + C_b \Vert \varphi - \phi \Vert^2\right) e^{\nu (t-r-\tau)}\,. \end{array}$$

Now, for t ∈[τ,τ + r], we deduce

$\begin{matrix} ∥ u_{t} - v_{t} ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} = \int_{- r}^{0} ∥ u (t + θ) - v (t + θ) ∥^{2} d θ, \\ \leq (r ∥ u^{0} - v^{0} ∥^{2} + (C_{b} r + 1) ∥ φ - ϕ ∥^{2}) e^{ν (t - r - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_t -v_t \Vert^2_{L^2([-r,0];L^2(\Omega))} = \int_{-r}^{0} \Vert u(t+\theta) - v(t+\theta) \Vert^2 d\theta\,,\\ \displaystyle \qquad\qquad\qquad\qquad\quad~\leq \left(r\Vert u^0-v^0 \Vert^2 + (C_br+1) \Vert \varphi - \phi \Vert^2\right) e^{\nu (t-r-\tau)}\,. \end{array}$$

So, for all t ≥ τ, we have

$\begin{matrix} ∥ u_{t} - v_{t} ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} \leq (r ∥ u^{0} - v^{0} ∥^{2} + (C_{b} r + 1) ∥ φ - ϕ ∥^{2}) e^{ν (t - r - τ)} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u_t -v_t \Vert^2_{L^2([-r,0];L^2(\Omega))} \leq \left(r\Vert u^0-v^0 \Vert^2 + (C_br+1) \Vert \varphi - \phi \Vert^2\right) e^{\nu (t-r-\tau)}\,. \end{array}$$

Hence, by this last estimate and (3.3) we deduce the continuity of S(t,τ).◼

3.3

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

$\begin{matrix} U (t, τ) D (τ) \subset B (t), \forall t - τ \geq T . \end{matrix}$ $$\begin{array}{} \displaystyle U(t,\tau)D(\tau) \subset B(t)\,,\; \forall t-\tau \geq T\,. \end{array}$$

On the other hand, we have

$\begin{matrix} S (t, τ) (u^{0}, φ) = j (U (t, τ) (u^{0}, φ)), \end{matrix}$ $$\begin{array}{} \displaystyle S(t,\tau)(u^0,\varphi) = j(U(t,\tau)(u^0,\varphi))\,, \end{array}$$

it follows that

$\begin{matrix} S (t, τ) (u^{0}, φ) = j (U (t, τ) (u^{0}, φ)) \subset j (B (t)), \forall t - τ \geq T . \end{matrix}$ $$\begin{array}{} \displaystyle S(t,\tau)(u^0,\varphi) = j(U(t,\tau)(u^0,\varphi)) \subset j(B(t))\,,\; \forall t-\tau \geq T\,. \end{array}$$

◼

Remark 2

Noticing that the word absorbing used in this papier should be interpreted in a generalized sense, since U is not a process.

Now, we need the following estimations.

Lemma 3

Assume that $\begin{matrix} g \in L_{l o c}^{2} (R; L^{2} (Ω)), \end{matrix}$ $\begin{array}{} \displaystyle g\in L^2_{loc}(\mathbb{R};L^2(\Omega))\,, \end{array}$there exists a small enough α < 2λ₁ – 2–C_bsuch that

$\begin{matrix} \int_{- \infty}^{t} e^{α t} ∥ g (s) ∥^{2} d s < \infty, \end{matrix}$ $$\begin{array}{} \displaystyle \int_{-\infty}^{t} e^{\alpha t} \Vert g(s) \Vert^2 ds \lt \infty \,, \end{array}$$(3.5)

the functionfsatisfies(1.2)-(1.5)andbfulfills conditions (I)-(IV) and

$\begin{matrix} \int_{τ}^{t} e^{σ s} ∥ b (s, u_{s}) - b (s, v_{s}) ∥^{2} d s \leq C_{b} \int_{τ - r}^{t} e^{σ s} ∥ u (s) - v (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau}^{t} e^{\sigma s} \Vert b(s,u_s) - b(s,v_s) \Vert^2 ds \leq C_b \int_{\tau -r}^{t} e^{\sigma s} \Vert u(s)-v(s) \Vert^2 ds\,. \end{array}$$(3.6)

Then we have

$\begin{matrix} ∥ u (t) ∥^{2} \leq e^{- α (t - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | α^{- 1} (1 - e^{- α (t - τ)}) + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u(t) \Vert^2 \leq e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds \nonumber\\ \displaystyle \qquad\quad\,\,+\,2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t-\tau)}\right) + e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$(3.7)

and

$\begin{matrix} η e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s + 2 μ_{1} e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ \leq e^{- α (t - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s + 2 c | Ω | α^{- 1} (1 - e^{- α (t - τ)}) \\ + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \quad\eta e^{-\alpha t}\int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds + 2\mu_1e^{-\alpha t} \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds \nonumber\\ \displaystyle \leq e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds + 2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t-\tau)}\right) \nonumber\\ \displaystyle +\, e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$(3.8)

where η : = 2λ₁ – 2 – α – C_b.

Proof

Multiplying (1.1) by u and integrating over Ω, one has

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} = \int_{Ω} f (u) u + \int_{Ω} b (t, u_{t}) u + \int_{Ω} g u . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2} \frac{d}{dt} \Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 = \int_{\Omega} f(u) u + \int_{\Omega} b(t,u_t) u + \int_{\Omega} g u\,. \end{array}$$

By (1.2), Cauchy-Shwarz and Young inequalities, we obtain

$\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} + μ_{1} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \leq c | Ω | + \frac{1}{2} ∥ b (t, u_{t}) ∥^{2} + \frac{1}{2} ∥ g (t) ∥^{2} + ∥ u (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{1}{2} \frac{d}{dt} \Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 + \mu_1 \Vert u(t) \Vert^p_{L^p(\Omega)} \leq c\vert \Omega \vert + \frac{1}{2} \Vert b(t,u_t)\Vert^2 + \frac{1}{2}\Vert g(t) \Vert^2 + \Vert u(t) \Vert^2\,. \end{array}$$

Since λ₁∥u∥² ≤ ∥∇u∥² and after calculation, one has

$\begin{matrix} \frac{d}{d t} ∥ u (t) ∥^{2} + 2 (λ_{1} - 1) ∥ u (t) ∥^{2} + 2 μ_{1} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \leq 2 c | Ω | + ∥ b (t, u_{t}) ∥^{2} + ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt} \Vert u(t) \Vert^2 + 2(\lambda_1 - 1)\Vert u(t) \Vert^2 + 2\mu_1 \Vert u(t) \Vert^p_{L^p(\Omega)} \leq 2 c\vert \Omega \vert + \Vert b(t,u_t)\Vert^2 + \Vert g(t) \Vert^2 \,. \end{array}$$

Now, we multiply this last estimate by e^αt such that 0 < α < 2λ₁ –2 – C_b, so one gets

$\begin{matrix} e^{α t} \frac{d}{d t} ∥ u (t) ∥^{2} + 2 (λ_{1} - 1) e^{α t} ∥ u (t) ∥^{2} + 2 μ_{1} e^{α t} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \\ \leq 2 c | Ω | e^{α t} + e^{α t} ∥ b (t, u_{t}) ∥^{2} + e^{α t} ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \quad \displaystyle e^{\alpha t} \frac{d}{dt} \Vert u(t) \Vert^2 + 2(\lambda_1 - 1)e^{\alpha t}\Vert u(t) \Vert^2 + 2\mu_1 e^{\alpha t}\Vert u(t) \Vert^p_{L^p(\Omega)} \\ \leq 2 c\vert \Omega \vert e^{\alpha t} + e^{\alpha t} \Vert b(t,u_t)\Vert^2 + e^{\alpha t} \Vert g(t) \Vert^2 \,. \end{array}$$(3.9)

On the other hand, we have

$\begin{matrix} \frac{d}{d t} (e^{α t} ∥ u (t) ∥^{2}) = α e^{α t} ∥ u (t) ∥^{2} + e^{α t} \frac{d}{d t} ∥ u (t) ∥^{2} \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left(e^{\alpha t}\Vert u(t) \Vert^2\right) = \alpha e^{\alpha t} \Vert u(t) \Vert^2 + e^{\alpha t} \frac{d}{dt} \Vert u(t) \Vert^2 \end{array}$$

We substitute (3.9) in this equality, we find

$\begin{matrix} \frac{d}{d t} (e^{α t} ∥ u (t) ∥^{2}) \leq α e^{α t} ∥ u (t) ∥^{2} - 2 (λ_{1} - 1) e^{α t} ∥ u (t) ∥^{2} - 2 μ_{1} e^{α t} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \\ + 2 c | Ω | e^{α t} + e^{α t} ∥ b (t, u_{t}) ∥^{2} + e^{α t} ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left(e^{\alpha t}\Vert u(t) \Vert^2\right) \leq \alpha e^{\alpha t} \Vert u(t) \Vert^2 -2 (\lambda_1 - 1)e^{\alpha t}\Vert u(t) \Vert^2 - 2\mu_1 e^{\alpha t}\Vert u(t) \Vert^p_{L^p(\Omega)} \\ \displaystyle \qquad\qquad\qquad\quad\,+ 2 c\vert \Omega \vert e^{\alpha t} + e^{\alpha t} \Vert b(t,u_t)\Vert^2 + e^{\alpha t} \Vert g(t) \Vert^2 \,. \end{array}$$

Integrating this last estimate over [τ, t], one obtains

$\begin{matrix} e^{α t} ∥ u (t) ∥^{2} \leq e^{α τ} ∥ u (τ) ∥^{2} + 2 c | Ω | α^{- 1} (e^{α t} - e^{α τ}) \\ + (α + 2 - 2 λ_{1}) \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s - 2 μ_{1} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ + \int_{τ}^{t} e^{α s} ∥ b (s, u_{s}) ∥^{2} d s + \int_{τ}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha t}\Vert u(t) \Vert^2 \leq e^{\alpha \tau}\Vert u(\tau) \Vert^2 + 2 c\vert \Omega \vert \alpha^{-1} (e^{\alpha t}- e^{\alpha \tau}) \\ \displaystyle \qquad\qquad\,\,\,\,+\, (\alpha +2-2 \lambda_1) \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds - 2\mu_1 \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds\\ \displaystyle \qquad\qquad\,\,\,\,+ \int_{\tau}^{t} e^{\alpha s} \Vert b(s,u_s)\Vert^2 ds + \int_{\tau}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

Using (3.6) and (II), one has

$\begin{matrix} e^{α t} ∥ u (t) ∥^{2} \leq e^{α τ} ∥ u (τ) ∥^{2} + 2 c | Ω | α^{- 1} (e^{α t} - e^{α τ}) \\ + (α + 2 - 2 λ_{1}) \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s - 2 μ_{1} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ + C_{b} \int_{τ - r}^{t} e^{α s} ∥ u (s) ∥^{2} d s + \int_{τ}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha t}\Vert u(t) \Vert^2 \leq e^{\alpha \tau}\Vert u(\tau) \Vert^2 + 2 c\vert \Omega \vert \alpha^{-1} (e^{\alpha t}- e^{\alpha \tau}) \nonumber\\ \displaystyle \qquad\qquad\,\,\,\,+\, (\alpha +2-2 \lambda_1) \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds - 2\mu_1 \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds \nonumber\\ \displaystyle \qquad\qquad\,\,\,\,+ \,C_b\int_{\tau-r}^{t} e^{\alpha s} \Vert u(s)\Vert^2 ds + \int_{\tau}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$(3.10)

On the other hand, we have

$\begin{matrix} \int_{τ - r}^{t} e^{α s} ∥ u (s) ∥^{2} d s = \int_{τ - r}^{τ} e^{α s} ∥ u (s) ∥^{2} d s + \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s, \\ \leq e^{α τ} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s + \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{\tau-r}^{t} e^{\alpha s} \Vert u(s)\Vert^2 ds = \int_{\tau-r}^{\tau} e^{\alpha s} \Vert u(s)\Vert^2 ds + \int_{\tau}^{t} e^{\alpha s} \Vert u(s)\Vert^2 ds \,,\nonumber\\ \displaystyle \qquad\qquad\qquad\qquad\leq e^{\alpha \tau}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds + \int_{\tau}^{t} e^{\alpha s} \Vert u(s)\Vert^2 ds \,. \end{array}$$(3.11)

So by (3.10) and (3.11), one finds

$\begin{matrix} e^{α t} ∥ u (t) ∥^{2} \leq e^{α τ} ∥ u (τ) ∥^{2} + 2 c | Ω | α^{- 1} (e^{α t} - e^{α τ}) \\ + (α + 2 - 2 λ_{1} + C_{b}) \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s - 2 μ_{1} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ + C_{b} e^{α τ} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s + \int_{τ}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha t}\Vert u(t) \Vert^2 \leq e^{\alpha \tau}\Vert u(\tau) \Vert^2 + 2 c\vert \Omega \vert \alpha^{-1} (e^{\alpha t}- e^{\alpha \tau}) \\ \displaystyle \qquad\qquad\,\,\,\,+\, (\alpha +2-2 \lambda_1 + C_b) \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds - 2\mu_1 \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds \\ \displaystyle \qquad\qquad\,\,\,\,+\, C_be^{\alpha \tau}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds + \int_{\tau}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

Hence, by (3.5) we obtain

$\begin{matrix} ∥ u (t) ∥^{2} + (2 λ_{1} - α - 2 - C_{b}) e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s \\ + 2 μ_{1} e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ \leq e^{- α (t - τ)} ∥ u (τ) ∥^{2} + 2 c | Ω | α^{- 1} (1 - e^{- α (t - τ)}) + C_{b} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \,\,\,\,\Vert u(t) \Vert^2 + (2 \lambda_1 - \alpha - 2 - C_b) e^{-\alpha t}\int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds \\ \displaystyle + 2\mu_1e^{-\alpha t} \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds \\ \displaystyle \leq e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + 2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t-\tau)}\right) + C_b e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds \\ \displaystyle + e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

Thus, for η : = 2 λ₁– α – 2 – C_b > 0, by this last estimate we get

$\begin{matrix} ∥ u (t) ∥^{2} \leq e^{- α (t - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | α^{- 1} (1 - e^{- α (t - τ)}) + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u(t) \Vert^2 \leq e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds \\ \displaystyle \qquad\quad\,\,+ 2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t-\tau)}\right) + e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$

and

$\begin{matrix} η e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥^{2} d s + 2 μ_{1} e^{- α t} \int_{τ}^{t} e^{α s} ∥ u (s) ∥_{L^{p} (Ω)}^{p} d s \\ \leq e^{- α (t - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s + 2 c | Ω | α^{- 1} (1 - e^{- α (t - τ)}) \\ + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \quad\eta e^{-\alpha t}\int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^2 ds + 2\mu_1e^{-\alpha t} \int_{\tau}^{t} e^{\alpha s}\Vert u(s) \Vert^p_{L^p(\Omega)} ds \\ \displaystyle \leq e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds + 2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t-\tau)}\right) \\ \displaystyle+ e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$

for all t ≥ τ. So by these two estimations the proof of the lemma is finished. ◼

Proposition 1

Under the assumptions inlemma (3). Then the family {B₁(t) : t ∈ ℝ} given by

$\begin{matrix} B_{1} (t) = {\bar{B}}_{C ([- r, 0]; L^{2} (Ω))} (0, R_{1} (t)), \end{matrix}$ $$\begin{array}{} \displaystyle B_1(t) = \overline{B}_{C([-r,0]; L^2(\Omega))}(0, R_1(t))\,, \end{array}$$

with

$\begin{matrix} R_{1}^{2} (t) = e^{α r} (2 c | Ω | α^{- 1} + e^{- α t} \int_{- \infty}^{t} e^{α t} ∥ g (s) ∥^{2} d s), \forall t \in R; \end{matrix}$ $$\begin{array}{} \displaystyle R_1^2(t)= e^{\alpha r} \left(2 c \vert \Omega \vert \alpha^{-1} + e^{-\alpha t} \int_{-\infty}^{t} e^{\alpha t} \Vert g(s) \Vert^2 ds \right) \,,\; \forall t \in \mathbb{R}\,; \end{array}$$

is pullback 𝓓-absorbing for the mappingU(t, τ). Moreover, the family {B₀(t) : t ∈ ℝ} given by

$\begin{matrix} B_{0} (t) = {\bar{B}}_{L^{2} (Ω))} (0, R_{1} (t)) \times {\bar{B}}_{L^{2} ([- r, 0]; L^{2} (Ω))} (0, \sqrt{r} R_{1} (t)) \subset H, \forall t \in R, \end{matrix}$ $$\begin{array}{} \displaystyle B_0(t) = \overline{B}_{L^2(\Omega))}(0, R_1(t)) \times \overline{B}_{L^2([-r,0]; L^2(\Omega))}\left(0, \sqrt{r}R_1(t)\right) \subset H\,,\; \forall t\in \mathbb{R}\,, \end{array}$$

is pullback 𝓓-absorbing for the process S defined by(3.1).

Proof

The first part may be proved as follows.

By definition, we have

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} = sup_{s \in [- r, 0]} ∥ u (t + s) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert U(t,\tau)(u^0,\varphi) \Vert_{C([-r,0];L^2(\Omega))}^2 = \sup_{s\in[-r,0]} \Vert u(t+s) \Vert^2 \,. \end{array}$$

From (3.7), if we take t ≥ τ + r, so t + θ ≥ τ. Then one has

$\begin{matrix} ∥ u (t + θ) ∥^{2} \leq e^{- α (t + θ - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t + θ - τ)} ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | α^{- 1} (1 - e^{- α (t + θ - τ)}) + e^{- α (t + θ)} \int_{- \infty}^{t + θ} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u(t+\theta) \Vert^2 \leq e^{-\alpha(t+\theta- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t+\theta-\tau)} \Vert \varphi\Vert_{L^2([-r,0];L^2(\Omega))}^2 \\ \displaystyle \qquad\qquad\quad+ 2 c\vert \Omega \vert \alpha^{-1} \left(1- e^{-\alpha (t+\theta-\tau)}\right) + e^{-\alpha (t+\theta)}\int_{-\infty}^{t+\theta} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$

which implies that

$\begin{matrix} sup_{s \in [- r, 0]} ∥ u (t + s) ∥^{2} \leq e^{- α (t - r - τ)} ∥ u (τ) ∥^{2} + C_{b} e^{- α (t - r - τ)} ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | α^{- 1} e^{α r} (e^{- α r} - e^{- α (t - τ)}) \\ + e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \end{matrix}$ $$\begin{array}{} \displaystyle \sup_{s\in [-r,0]}\Vert u(t+s) \Vert^2 \leq e^{-\alpha(t-r- \tau)}\Vert u(\tau) \Vert^2 + C_b e^{-\alpha (t-r-\tau)}\Vert \varphi\Vert_{L^2([-r,0];L^2(\Omega))}^2 \nonumber\\ \displaystyle \qquad\qquad\qquad\qquad+ \,2 c\vert \Omega \vert \alpha^{-1} e^{\alpha r}\left(e^{-\alpha r}- e^{-\alpha (t-\tau)}\right) \nonumber\\ \displaystyle \qquad\qquad\qquad\qquad+\, e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,, \end{array}$$(3.12)

On the one hand, we have

$\begin{matrix} ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} = \int_{- r}^{0} ∥ φ (θ) ∥^{2} d θ, \\ \leq \int_{- r}^{0} sup_{s \in [- r, 0]} ∥ φ (s) ∥^{2} d θ, \\ \leq r ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert \varphi\Vert_{L^2([-r,0];L^2(\Omega))}^2 = \int_{-r}^{0} \Vert \varphi(\theta)\Vert^2 d\theta \,,\nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\leq \int_{-r}^{0} \sup_{s\in [-r,0]}\Vert \varphi(s)\Vert^2 d\theta\,,\nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\leq r \Vert \varphi \Vert^2_{C([-r,0];L^2(\Omega))}\,. \end{array}$$(3.13)

Therefore by (3.12), (3.13) and the fact that u(τ) = φ (0), we obtain

$\begin{matrix} sup_{s \in [- r, 0]} ∥ u (t + s) ∥^{2} \leq e^{- α (t - r - τ)} ∥ φ (0) ∥^{2} + C_{b} r e^{- α (t - r - τ)} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | α^{- 1} e^{α r} (e^{- α r} - e^{- α (t - τ)}) + e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \\ \leq (1 + C_{b} r) e^{- α (t - r - τ)} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} + 2 c | Ω | α^{- 1} e^{α r} \\ + e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\sup_{s\in [-r,0]}\Vert u(t+s) \Vert^2 \leq e^{-\alpha(t-r- \tau)}\Vert \varphi(0) \Vert^2 + C_b r e^{-\alpha (t-r-\tau)}\Vert \varphi\Vert_{C([-r,0];L^2(\Omega))}^2 \\ \displaystyle +\, 2 c\vert \Omega \vert \alpha^{-1} e^{\alpha r}\left(e^{-\alpha r}- e^{-\alpha (t-\tau)}\right) + e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,,\\ \displaystyle \leq (1+C_b r) e^{-\alpha (t-r-\tau)} \Vert \varphi\Vert_{C([-r,0];L^2(\Omega))}^2 + 2 c\vert \Omega \vert \alpha^{-1} e^{\alpha r} \\ \displaystyle + \,e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

Then, we find

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} = sup_{s \in [- r, 0]} ∥ u (t + s) ∥^{2}, \\ \leq (1 + C_{b} r) e^{- α (t - r - τ)} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + e^{α r} (2 c | Ω | α^{- 1} + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s), \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert U(t,\tau)(u^0,\varphi) \Vert_{C([-r,0];L^2(\Omega))}^2 = \sup_{s\in [-r,0]}\Vert u(t+s) \Vert^2 \,,\nonumber\\ \leq (1+C_b r) e^{-\alpha (t-r-\tau)} \Vert \varphi\Vert_{C([-r,0];L^2(\Omega))}^2 \nonumber\\ +\, e^{\alpha r} \left(2 c\vert \Omega \vert \alpha^{-1} + e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds\right) \,, \end{array}$$(3.14)

for all (u⁰, φ) ∈ H and all t ≥ τ + r.

Let 𝓡 be the set of all functions ρ : ℝ ⟶ (0, + ∞) such that

$\begin{matrix} lim_{t \to - \infty} e^{α t} ρ^{2} (t) = 0 . \end{matrix}$ $$\begin{array}{} \displaystyle \lim_{t\rightarrow -\infty} e^{\alpha t} \rho^{2}(t) =0\,. \end{array}$$

By 𝓓 we denote the class of all families D̂ = {D(t) : t ∈ ℝ} ⊂ 𝓟(C([–r, 0];L²(Ω))) such that D(t) ⊂ B_{C([–r, 0];L²(Ω))}(0,ρ(t)), for some ρ ∈ 𝓡, where we denote by B_{C([–r, 0];L²(Ω))}(0, ρ(t)) the closed ball in C([–r, 0];L²(Ω)) centered at 0 with radius ρ(t). Let

$\begin{matrix} R_{1}^{2} (t) = e^{α r} (2 c | Ω | α^{- 1} + e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s) . \end{matrix}$ $$\begin{array}{} \displaystyle R_1^2(t)= e^{\alpha r} \left(2 c\vert \Omega \vert \alpha^{-1} + e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds\right)\,. \end{array}$$

Thus, for all D̂ ∈ 𝓓 and all t ∈ ℝ, by (3.14) there exists τ₀ (D̂, t) ≤ t such that

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \leq R_{1}^{2} (t), \end{matrix}$ $$\begin{array}{} \displaystyle \Vert U(t,\tau)(u^0,\varphi) \Vert_{C([-r,0];L^2(\Omega))}^2 \leq R_1^2(t)\,, \end{array}$$(3.15)

for all τ ≤ τ₀(D̂, t); i.e., B₁(t) = B_{C([–r, 0]; L²(Ω))}(0, R₁(t)) is pullback 𝓓-absorbing for the mapping U(t, τ).

Concerning the second part, we observe that {j(B(t)), t ∈ ℝ} is a family of pullback 𝓓-absorbing sets for the process S. On the other hand, since

$\begin{matrix} ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} \leq r ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2}, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert \varphi\Vert_{L^2([-r,0];L^2(\Omega))}^2 \leq r \Vert \varphi\Vert_{C([-r,0];L^2(\Omega))}^2 \,, \end{array}$$

and

$\begin{matrix} j (B (t)) = ((φ (0), φ) : φ \in {\bar{B}}_{C ([- r, 0]; L^{2} (Ω))} (0, R_{1} (t))), \end{matrix}$ $$\begin{array}{} \displaystyle j(B(t)) = \left\{(\varphi(0),\varphi)\,:\; \varphi \in \overline{B}_{C([-r,0]; L^2(\Omega))}(0, R_1(t))\right\}\,, \end{array}$$

we deduce that

$\begin{matrix} j (B (t)) \subset {\bar{B}}_{L^{2} (Ω))} (0, R_{1} (t)) \times {\bar{B}}_{L^{2} ([- r, 0]; L^{2} (Ω))} (0, \sqrt{r} R_{1} (t)) = B_{0} (t), \end{matrix}$ $$\begin{array}{} \displaystyle j(B(t)) \subset \overline{B}_{L^2(\Omega))}(0, R_1(t)) \times \overline{B}_{L^2([-r,0]; L^2(\Omega))}\left(0, \sqrt{r}R_1(t)\right) = B_0(t)\,, \end{array}$$

which implies that the family {B₀(t): t ∈ ℝ} is pullback 𝓓-absorbing sets for the process S. ◼

3.4

Existence of pullback D-absorbing set in

\begin{matrix} C ([- r, 0]; H_{0}^{1} (Ω)) \end{matrix}

$\begin{array}{} \displaystyle C([-r,0]; H^1_0(\Omega)) \end{array}$

Proposition 2

Suppose that conditions oflemma (3)are satisfied, if there exists a sufficiently smallα^∗such that

$\begin{matrix} α < α^{*} < min (2 \frac{λ_{1} - 1}{λ_{1}}, 2 μ_{1}) . \end{matrix}$ $$\begin{array}{} \displaystyle \alpha \lt \alpha^* \lt \min \left\{ 2 \frac{\lambda_1 -1}{\lambda_1}\,,\, 2\mu_1 \right\}\,. \end{array}$$

Then the family {B₂(t) : t ∈ ℝ} given by

$\begin{matrix} B_{2} (t) = {\bar{B}}_{C ([- r, 0]; H_{0}^{1} (Ω))} (0, R_{2} (t)), \end{matrix}$ $$\begin{array}{} \displaystyle B_2(t) = \overline{B}_{C([-r,0]; H^1_0(\Omega))}(0, R_2(t))\,, \end{array}$$

where

$\begin{matrix} R_{2}^{2} (t) = 2 c | Ω | ({α^{*}}^{- 1} e^{α^{*} r} + 2 C_{b} α^{- 1} η^{- 1} e^{α r}) \\ + 2 C_{b} η^{- 1} e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s + 2 e^{- α^{*} (t - r)} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s, \forall t \in R, \end{matrix}$ $$\begin{array}{} \displaystyle R_2^2(t)= 2c \vert \Omega \vert \left({\alpha^*}^{-1} e^{\alpha^* r }+ 2C_b \alpha^{-1} \eta^{-1} e^{\alpha r}\right) \\ \displaystyle \qquad\,\,\,\,+\, 2C_b \eta^{-1} e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds + 2 e^{-\alpha^* (t-r)} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \,,\; \forall t\in \mathbb{R}\,, \end{array}$$

is pullback 𝓓-absorbing for the mapping U(t, τ).

Proof

Multipying (1.1) by $\begin{matrix} u + \frac{\partial u}{\partial t} \end{matrix}$ $\begin{array}{} \displaystyle u+\frac{\partial u}{\partial t} \end{array}$ and integrating over Ω, we obtain

$\begin{matrix} {(\frac{d}{d t} u (t))}^{2} + \frac{1}{2} \frac{d}{d t} (∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2}) \\ = \int_{Ω} f (u) (u + \frac{\partial u}{\partial t}) + \int_{Ω} b (t, u_{t}) (u + \frac{\partial u}{\partial t}) + \int_{Ω} g (u + \frac{\partial u}{\partial t}) . \end{matrix}$ $$\begin{array}{} \displaystyle \quad \left\Vert \frac{d}{dt} u(t) \right\Vert^2 + \frac{1}{2} \frac{d}{dt} \left(\Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2\right) \\ \displaystyle = \int_{\Omega} f(u)\left( u+\frac{\partial u}{\partial t} \right) + \int_{\Omega} b(t,u_t) \left( u+\frac{\partial u}{\partial t} \right) + \int_{\Omega} g\left( u+\frac{\partial u}{\partial t} \right)\,. \end{array}$$

Using (1.2), (1.5), (2.5) and Young inequality, one finds

$\begin{matrix} 2 {(\frac{d}{d t} u (t))}^{2} + \frac{d}{d t} (∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} + 2 μ_{1}^{'} ∥ u (t) ∥_{L^{p} (Ω)}^{p}) \\ + 2 ∥ \nabla u (t) ∥^{2} + 2 μ_{1} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \\ \leq 2 c | Ω | + 2 ∥ b (t, u_{t}) ∥^{2} + 2 ∥ g (t) ∥^{2} + 2 {(\frac{d}{d t} u (t))}^{2} + 2 ∥ u (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \,\,\,\,\,2\left\Vert \frac{d}{dt} u(t) \right\Vert^2 + \frac{d}{dt} \left(\Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 + 2\mu'_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \right) \\ \displaystyle + 2 \Vert \nabla u(t) \Vert^2 + 2\mu_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \\ \displaystyle \leq 2c \vert\Omega \vert + 2\Vert b(t,u_t) \Vert^2 + 2\Vert g(t) \Vert^2 + 2\left\Vert \frac{d}{dt} u(t) \right\Vert^2 + 2\Vert u(t) \Vert^2\,. \end{array}$$

By the fact that λ₁∥u∥² ≤ ∥∇u∥², after simplification one has

$\begin{matrix} \frac{d}{d t} (∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} + 2 μ_{1}^{'} ∥ u (t) ∥_{L^{p} (Ω)}^{p}) \\ + 2 (1 - λ_{1}^{- 1}) ∥ \nabla u (t) ∥^{2} + 2 μ_{1} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \\ \leq 2 c | Ω | + 2 ∥ b (t, u_{t}) ∥^{2} + 2 ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\frac{d}{dt} \left(\Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 + 2\mu'_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \right) \\ \displaystyle + 2(1-\lambda_1^{-1}) \Vert \nabla u(t) \Vert^2 + 2\mu_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \\ \displaystyle \leq 2c \vert\Omega \vert + 2\Vert b(t,u_t) \Vert^2 + 2\Vert g(t) \Vert^2 \,. \end{array}$$

Since α in lemma (3) is small enough, we can choose a positive constant α^∗ sufficiently small with α < α^* < $\begin{matrix} min (2 \frac{λ_{1} - 1}{λ_{1}}, 2 μ_{1}), \end{matrix}$ $\begin{array}{} \displaystyle \min \left\{ 2 \frac{\lambda_1 -1}{\lambda_1}\,,\, 2\mu_1 \right\}\,, \end{array}$ such that

$\begin{matrix} 2 (1 - λ_{1}^{- 1}) ∥ \nabla u (t) ∥^{2} \geq α^{*} (∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2}) . \end{matrix}$ $$\begin{array}{} \displaystyle 2(1-\lambda_1^{-1}) \Vert \nabla u(t) \Vert^2 \geq \alpha^* (\Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2)\,. \end{array}$$

So, we can write

$\begin{matrix} \frac{d}{d t} γ_{1} (t) + α^{*} γ_{1} (t) \leq 2 c | Ω | + 2 ∥ b (t, u_{t}) ∥^{2} + 2 ∥ g (t) ∥^{2}, \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\gamma_1 (t) + \alpha^* \gamma_1 (t) \leq 2c \vert \Omega \vert + 2 \Vert b(t,u_t) \Vert^2 + 2 \Vert g(t) \Vert^2 \,, \end{array}$$(3.16)

where

$\begin{matrix} γ_{1} (t) = ∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} + 2 μ_{1}^{'} ∥ u (t) ∥_{L^{p} (Ω)}^{p} . \end{matrix}$ $$\begin{array}{} \displaystyle \gamma_1 (t) = \Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 + 2\mu'_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \,. \end{array}$$(3.17)

Multiplying (3.16) by e^{α^∗t}, one has

$\begin{matrix} e^{α^{*} t} \frac{d}{d t} γ_{1} (t) + α^{*} e^{α^{*} t} γ_{1} (t) \leq 2 c | Ω | e^{α^{*} t} + 2 e^{α^{*} t} ∥ b (t, u_{t}) ∥^{2} + 2 e^{α^{*} t} ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha^* t} \frac{d}{dt}\gamma_1 (t) + \alpha^* e^{\alpha^* t} \gamma_1 (t) \leq 2c \vert \Omega \vert e^{\alpha^* t} + 2 e^{\alpha^* t} \Vert b(t,u_t) \Vert^2+ 2 e^{\alpha^* t}\Vert g(t) \Vert^2 \,. \end{array}$$(3.18)

On the other hand, we have

$\begin{matrix} \frac{d}{d t} (e^{α^{*} t} γ_{1} (t)) = α^{*} e^{α^{*} t} γ_{1} (t) + e^{α^{*} t} \frac{d}{d t} γ_{1} (t) \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left(e^{\alpha^* t} \gamma_1 (t)\right) = \alpha^* e^{\alpha^* t} \gamma_1 (t) + e^{\alpha^* t} \frac{d}{dt}\gamma_1 (t) \end{array}$$(3.19)

Then, by (3.18) and (3.19), we obtain

$\begin{matrix} \frac{d}{d t} (e^{α^{*} t} γ_{1} (t)) \leq α^{*} e^{α^{*} t} γ_{1} (t) - α^{*} e^{α^{*} t} γ_{1} (t) + 2 c | Ω | e^{α^{*} t} \\ + 2 e^{α^{*} t} ∥ b (t, u_{t}) ∥^{2} + 2 e^{α^{*} t} ∥ g (t) ∥^{2}, \\ \leq 2 c | Ω | e^{α^{*} t} + 2 e^{α^{*} t} ∥ b (t, u_{t}) ∥^{2} + 2 e^{α^{*} t} ∥ g (t) ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \frac{d}{dt}\left(e^{\alpha^* t} \gamma_1 (t)\right) \leq \alpha^* e^{\alpha^* t} \gamma_1 (t) - \alpha^* e^{\alpha^* t} \gamma_1 (t) + 2c \vert \Omega \vert e^{\alpha^* t} \\ \displaystyle \qquad\qquad\qquad\,\,\,+\,2 e^{\alpha^* t} \Vert b(t,u_t) \Vert^2+ 2 e^{\alpha^* t}\Vert g(t) \Vert^2 \,,\\ \displaystyle \qquad\qquad\qquad\,\,\,\leq 2c \vert \Omega \vert e^{\alpha^* t} + 2 e^{\alpha^* t} \Vert b(t,u_t) \Vert^2 + 2 e^{\alpha^* t}\Vert g(t) \Vert^2 \,. \end{array}$$

Integrating this last one from τ to t, one gets

$\begin{matrix} e^{α^{*} t} γ_{1} (t) \leq e^{α^{*} τ} γ_{1} (τ) + 2 c | Ω | \int_{τ}^{t} e^{α^{*} s} d s + 2 \int_{τ}^{t} e^{α^{*} s} ∥ b (s, u_{s}) ∥^{2} d s \\ + 2 \int_{τ}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s, \\ + \leq e^{α^{*} τ} γ_{1} (τ) + 2 c | Ω | {α^{*}}^{- 1} (e^{α^{*} t} - e^{α^{*} τ}) + 2 \int_{τ}^{t} e^{α^{*} s} ∥ b (s, u_{s}) ∥^{2} d s \\ + + 2 \int_{τ}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha^* t} \gamma_1 (t) \leq e^{\alpha^* \tau} \gamma_1 (\tau) + 2c \vert \Omega \vert \int_{\tau}^{t}e^{\alpha^* s} ds+ 2 \int_{\tau}^{t} e^{\alpha^* s} \Vert b(s,u_s) \Vert^2 ds\\ \displaystyle \qquad\qquad+\, 2 \int_{\tau}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,,\\ \displaystyle \qquad\qquad+\,\leq e^{\alpha^* \tau} \gamma_1 (\tau) + 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( e^{\alpha^* t} - e^{\alpha^* \tau}\right) + 2 \int_{\tau}^{t} e^{\alpha^* s} \Vert b(s,u_s) \Vert^2 ds\\ \displaystyle \qquad\qquad+\,+\, 2 \int_{\tau}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,. \end{array}$$

From (3.5) and (3.6), one finds

$\begin{matrix} e^{α^{*} t} γ_{1} (t) \leq e^{α^{*} τ} γ_{1} (τ) + 2 c | Ω | {α^{*}}^{- 1} (e^{α^{*} t} - e^{α^{*} τ}) + 2 C_{b} \int_{τ - r}^{t} e^{α^{*} s} ∥ u (s) ∥^{2} d s \\ + 2 \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s, \\ \leq e^{α^{*} τ} γ_{1} (τ) + 2 c | Ω | {α^{*}}^{- 1} (e^{α^{*} t} - e^{α^{*} τ}) + 2 C_{b} e^{α^{*} τ} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 C_{b} \int_{τ}^{t} e^{α^{*} s} ∥ u (s) ∥^{2} d s + 2 \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle e^{\alpha^* t} \gamma_1 (t) \leq e^{\alpha^* \tau} \gamma_1 (\tau) + 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( e^{\alpha^* t} - e^{\alpha^* \tau}\right) + 2C_b \int_{\tau-r}^{t} e^{\alpha^* s} \Vert u(s) \Vert^2 ds\\ \displaystyle \qquad\qquad+\, 2 \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,,\\ \displaystyle \qquad\qquad\leq e^{\alpha^* \tau} \gamma_1 (\tau) + 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( e^{\alpha^* t} - e^{\alpha^* \tau}\right) + 2C_b e^{\alpha^* \tau}\int_{\tau-r}^{\tau} \Vert u(s) \Vert^2 ds\\ \displaystyle \qquad\qquad+\, 2C_b \int_{\tau}^{t} e^{\alpha^* s} \Vert u(s) \Vert^2 ds + 2 \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,. \end{array}$$

We multiply this estimate by e^{–α^∗t}, we obtain

$\begin{matrix} γ_{1} (t) \leq e^{- α^{*} (t - τ)} γ_{1} (τ) + 2 C_{b} e^{- α^{*} (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) + 2 C_{b} e^{- α^{*} t} \int_{τ}^{t} e^{α^{*} s} ∥ u (s) ∥^{2} d s \\ + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \gamma_1(t) \leq e^{-\alpha^* (t-\tau)} \gamma_1 (\tau) +2C_b e^{-\alpha^* (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s) \Vert^2 ds \nonumber\\ \displaystyle \qquad\,\,+\,2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) + 2C_b e^{-\alpha^* t} \int_{\tau}^{t} e^{\alpha^* s} \Vert u(s) \Vert^2 ds \nonumber\\ \displaystyle \qquad\,\,+\,2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,. \end{array}$$(3.20)

On the one hand, since $\begin{matrix} H_{0}^{1} (Ω) \subset L^{2} (Ω) a n d H_{0}^{1} (Ω) \subset L^{p} (Ω), \end{matrix}$ $\begin{array}{} \displaystyle H^1_0(\Omega) \subset L^2(\Omega)\, {\rm and}\, H^1_0(\Omega) \subset L^p(\Omega)\,, \end{array}$we have

$\begin{matrix} γ_{1} (τ) = ∥ u (τ) ∥^{2} + ∥ \nabla u (τ) ∥^{2} + 2 μ_{1}^{'} ∥ u (τ) ∥_{L^{p} (Ω)}^{p}, \\ \leq (1 + λ_{1}^{- 1}) ∥ \nabla u (τ) ∥^{2} + 2 μ_{1}^{'} ∥ u (τ) ∥_{L^{p} (Ω)}^{p}, \\ \leq (1 + λ_{1}^{- 1}) ∥ \nabla u (τ) ∥^{2} + k_{1} ∥ \nabla u (τ) ∥^{p}, \\ \leq k_{2} (1 + λ_{1}^{- 1}) ∥ \nabla u (τ) ∥^{p} + k_{1} ∥ \nabla u (τ) ∥^{p}, \\ \leq k_{3} ∥ \nabla u (τ) ∥^{p} . \end{matrix}$ $$\begin{array}{} \displaystyle \gamma_1 (\tau) = \Vert u(\tau) \Vert^2 + \Vert \nabla u(\tau) \Vert^2 + 2\mu'_1 \Vert u(\tau) \Vert_{L^p(\Omega)}^p \,,\nonumber\\ \displaystyle \qquad\,\,\,\leq (1+ \lambda_1^{-1}) \Vert \nabla u(\tau) \Vert^2 + 2\mu'_1 \Vert u(\tau) \Vert_{L^p(\Omega)}^p \,,\nonumber\\ \displaystyle \qquad\,\,\, \leq (1+ \lambda_1^{-1}) \Vert \nabla u(\tau) \Vert^2 + k_1 \Vert \nabla u(\tau) \Vert^p\,,\nonumber\\ \displaystyle \qquad\,\,\, \leq k_2 (1+ \lambda_1^{-1}) \Vert \nabla u(\tau) \Vert^p + k_1 \Vert \nabla u(\tau) \Vert^p \,,\nonumber\\ \displaystyle \qquad\,\,\,\leq k_3 \Vert \nabla u(\tau) \Vert^p\,. \end{array}$$(3.21)

So, by (3.17), (3.20) and (3.21), one finds

$\begin{matrix} ∥ u (t) ∥^{2} + ∥ \nabla u (t) ∥^{2} + 2 μ_{1}^{'} ∥ u (t) ∥_{L^{p} (Ω)}^{p} \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} \\ + 2 C_{b} e^{- α^{*} (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) \\ + 2 C_{b} e^{- α^{*} t} \int_{τ}^{t} e^{α^{*} s} ∥ u (s) ∥^{2} d s + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \,\,\,\,\,\Vert u(t) \Vert^2 + \Vert \nabla u(t) \Vert^2 + 2\mu'_1 \Vert u(t) \Vert_{L^p(\Omega)}^p \leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p \\ \displaystyle +2C_b e^{-\alpha^* (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s) \Vert^2 ds + 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) \\ \displaystyle + 2C_b e^{-\alpha^* t} \int_{\tau}^{t} e^{\alpha^* s} \Vert u(s) \Vert^2 ds + 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds\,. \end{array}$$

From this last estimate and (3.8), we have

$\begin{matrix} ∥ \nabla u (t) ∥^{2} \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} e^{- α^{*} (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s \\ + 2 C_{b} e^{- α^{*} t} \int_{τ}^{t} e^{α^{*} s} ∥ u (s) ∥^{2} d s, \\ \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} e^{- α^{*} (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s \\ + 2 C_{b} η^{- 1} e^{- α (t - τ)} ∥ u (τ) ∥^{2} + 2 C_{b}^{2} η^{- 1} e^{- α (t - τ)} \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 4 C_{b} c | Ω | α^{- 1} η^{- 1} (1 - e^{- α (t - τ)}) + 2 C_{b} η^{- 1} e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s, \\ \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} (e^{- α^{*} (t - τ)} + C_{b} η^{- 1} e^{- α (t - τ)}) \int_{τ - r}^{τ} ∥ u (s) ∥^{2} d s \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) + 4 C_{b} c | Ω | α^{- 1} η^{- 1} (1 - e^{- α (t - τ)}) \\ + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert \nabla u(t) \Vert^2 \leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b e^{-\alpha^* (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s) \Vert^2 ds \\ \displaystyle \qquad\qquad\,+\, 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) + 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \\ \displaystyle \qquad\qquad\, +\, 2C_b e^{-\alpha^* t} \int_{\tau}^{t} e^{\alpha^* s} \Vert u(s) \Vert^2 ds \,,\\ \displaystyle \qquad\qquad\, \leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b e^{-\alpha^* (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s) \Vert^2 ds \\ \displaystyle \qquad\qquad\, + \,2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) + 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \\ \displaystyle \qquad\qquad\, +\, 2C_b \eta^{-1} e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 + 2C^2_b \eta^{-1} e^{-\alpha (t-\tau)}\int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds \\ \displaystyle \qquad\qquad\,+\, 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (t-\tau)}\right) + 2C_b \eta^{-1} e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,,\\ \displaystyle \qquad\qquad\,\leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2\\ \displaystyle \qquad\qquad\,+\, 2C_b \left(e^{-\alpha^* (t-\tau)} + C_b \eta^{-1} e^{-\alpha (t-\tau)} \right) \int_{\tau-r}^{\tau} \Vert u(s)\Vert^2 ds \\ \displaystyle \qquad\qquad\,+\,2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (t-\tau)}\right) \\ \displaystyle \qquad\qquad\,+\,2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

In the fact that

$\begin{matrix} ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} \leq r ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2}, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert \varphi \Vert_{L^2([-r,0]; L^2(\Omega))}^2 \leq r \Vert \varphi \Vert_{C([-r,0]; L^2(\Omega))}^2\,, \end{array}$$

one has

$\begin{matrix} ∥ \nabla u (t) ∥^{2} \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t - τ)} + C_{b} η^{- 1} e^{- α (t - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - τ)}) + 4 C_{b} c | Ω | α^{- 1} η^{- 1} (1 - e^{- α (t - τ)}) \\ + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s \\ \leq k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t - τ)} + C_{b} η^{- 1} e^{- α (t - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | {α^{*}}^{- 1} + 4 C_{b} c | Ω | α^{- 1} η^{- 1} \\ + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \quad\displaystyle \Vert \nabla u(t) \Vert^2 \leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 \nonumber\\ \displaystyle + 2C_br \left(e^{-\alpha^* (t-\tau)} + C_b \eta^{-1} e^{-\alpha (t-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \nonumber\\ \displaystyle + 2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-\tau)}\right) + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (t-\tau)}\right) \nonumber\\ \displaystyle + 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,\nonumber\\ \displaystyle \leq k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 \nonumber\\ + 2C_br \left(e^{-\alpha^* (t-\tau)} + C_b \eta^{-1} e^{-\alpha (t-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \nonumber\\ \displaystyle + 2c \vert \Omega \vert {\alpha^*}^{-1} + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \nonumber\\ \displaystyle + 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds\,. \end{array}$$(3.22)

If we take t ≥ τ + r i.e. t + θ ≥ τ, it follows

$\begin{matrix} ∥ \nabla u (t + θ) ∥^{2} \leq k_{3} e^{- α^{*} (t + θ - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t + θ - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t + θ - τ)} + C_{b} η^{- 1} e^{- α (t + θ - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t + θ - τ)}) + 4 C_{b} c | Ω | α^{- 1} η^{- 1} (1 - e^{- α (t + θ - τ)}) \\ + 2 e^{- α^{*} (t + θ)} \int_{- \infty}^{t + θ} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α (t + θ)} \int_{- \infty}^{t + θ} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert \nabla u(t+\theta) \Vert^2 \leq k_3 e^{-\alpha^* (t+\theta-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t+\theta- \tau)}\Vert u(\tau) \Vert^2 \\ \displaystyle +\,2C_br \left(e^{-\alpha^* (t+\theta-\tau)} + C_b \eta^{-1} e^{-\alpha (t+\theta-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \\ \displaystyle +\,2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t+\theta-\tau)}\right)+ 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (t+\theta-\tau)}\right) \\ \displaystyle +\,2 e^{-\alpha^* (t+\theta)} \int_{-\infty}^{t+\theta} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha (t+\theta)}\int_{-\infty}^{t+\theta} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

Hence,

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; H_{0}^{1} (Ω))}^{2} = sup_{θ \in [- r, 0]} ∥ \nabla u (t + θ) ∥^{2}, \\ \leq k_{3} e^{- α^{*} (t - r - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - r - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t - r - τ)} + C_{b} η^{- 1} e^{- α (t - r - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | {α^{*}}^{- 1} (1 - e^{- α^{*} (t - r - τ)}) + 4 C_{b} c | Ω | α^{- 1} η^{- 1} (1 - e^{- α (t - r - τ)}) \\ + 2 e^{- α^{*} (t - r)} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert U(t,\tau)(u^0,\varphi) \Vert^2_{C([-r,0];H^1_0(\Omega))} = \sup_{\theta\in[-r,0]} \Vert \nabla u(t+\theta) \Vert^2\,,\\ \displaystyle \leq k_3 e^{-\alpha^* (t-r-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t-r- \tau)}\Vert u(\tau) \Vert^2 \\ \displaystyle +\,2C_br \left(e^{-\alpha^* (t-r-\tau)} + C_b \eta^{-1} e^{-\alpha (t-r-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \\ \displaystyle +\,2c \vert \Omega \vert {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (t-r-\tau)}\right)+ 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (t-r-\tau)}\right) \\ \displaystyle +\,2 e^{-\alpha^* (t-r)} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \,. \end{array}$$

So, we obtain

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; H_{0}^{1} (Ω))}^{2} \\ \leq k_{3} e^{- α^{*} (t - r - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - r - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t - r - τ)} + C_{b} η^{- 1} e^{- α (t - r - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | ({α^{*}}^{- 1} e^{α^{*} r} + 2 C_{b} α^{- 1} η^{- 1} e^{α r}) \\ + 2 C_{b} η^{- 1} e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s \\ + 2 e^{- α^{*} (t - r)} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert U(t,\tau)(u^0,\varphi) \Vert^2_{C([-r,0];H^1_0(\Omega))} \nonumber\\ \displaystyle \leq k_3 e^{-\alpha^* (t-r-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t-r- \tau)}\Vert u(\tau) \Vert^2 \nonumber\\ \displaystyle +\,2C_br \left(e^{-\alpha^* (t-r-\tau)} + C_b \eta^{-1} e^{-\alpha (t-r-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \nonumber\\ \displaystyle +\, 2c \vert \Omega \vert \left({\alpha^*}^{-1} e^{\alpha^* r }+ 2C_b \alpha^{-1} \eta^{-1} e^{\alpha r}\right) \nonumber\\ \displaystyle + \,2C_b \eta^{-1} e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \nonumber\\ \displaystyle + \,2 e^{-\alpha^* (t-r)} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \,. \end{array}$$(3.23)

Similarly to the Lemma 3, let 𝓡 be the set of all functions ρ : ℝ ⟶ (0, + ∞) such that

$\begin{matrix} lim_{t \to - \infty} e^{α^{*} t} ρ^{2} (t) = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \lim_{t\rightarrow -\infty} e^{\alpha^* t} \rho^{2}(t) =0\,, \end{array}$$

by 𝓓 we denote the class of all families $\begin{matrix} \hat{D} = {D (t) : t \in R} \subset P (C ([- r, 0]; H_{0}^{1} (Ω))) \end{matrix}$ $\begin{array}{} \displaystyle \mathbf{\widehat{D}} = \{D(t) : t\in \mathbb{R} \} \subset \mathcal{P}(C([-r,0];H^1_0(\Omega))) \end{array}$ such that D(t) ⊂ $\begin{matrix} {\bar{B}}_{C ([- r, 0]; H_{0}^{1} (Ω))} (0, ρ (t)), \end{matrix}$ $\begin{array}{} \displaystyle \mathbf{\overline{B}}_{C([-r,0];H^1_0(\Omega))}(0,\rho(t))\,, \end{array}$ for some ρ ∈ 𝓡, where we denote by $\begin{matrix} {\bar{B}}_{C ([- r, 0]; H_{0}^{1} (Ω))} (0, ρ (t)) \end{matrix}$ $\begin{array}{} \displaystyle \mathbf{\overline{B}}_{C([-r,0];H^1_0(\Omega))}(0,\rho(t)) \end{array}$ the closed ball in $\begin{matrix} C ([- r, 0]; H_{0}^{1} (Ω)) \end{matrix}$ $\begin{array}{} \displaystyle C([-r,0];H^1_0(\Omega)) \end{array}$ centered at 0 with radius ρ(t). Let

$\begin{matrix} R_{2}^{2} (t) = 2 c | Ω | ({α^{*}}^{- 1} e^{α^{*} r} + 2 C_{b} α^{- 1} η^{- 1} e^{α r}) \\ + 2 C_{b} η^{- 1} e^{- α (t - r)} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s + 2 e^{- α^{*} (t - r)} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle R_2^2(t)= 2c \vert \Omega \vert \left({\alpha^*}^{-1} e^{\alpha^* r }+ 2C_b \alpha^{-1} \eta^{-1} e^{\alpha r}\right) \\ \displaystyle \qquad\,\,\,\,+\,2C_b \eta^{-1} e^{-\alpha (t-r)}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds + 2 e^{-\alpha^* (t-r)} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \,. \end{array}$$

Thus, for all D̂ ∈ 𝓓 and all t ∈ ℝ, by (3.23) there exists τ₀ (D̂, t) ≤ t such that

$\begin{matrix} ∥ U (t, τ) (u^{0}, φ) ∥_{C ([- r, 0]; H_{0}^{1} (Ω))}^{2} \leq R_{2}^{2} (t), \end{matrix}$ $$\begin{array}{} \displaystyle \Vert U(t,\tau)(u^0,\varphi) \Vert_{C([-r,0];H^1_0(\Omega))}^2 \leq R_2^2(t)\,, \end{array}$$(3.24)

for all τ ≤ τ₀(D̂, t), this means that $\begin{matrix} B_{2} (t) = {\bar{B}}_{C ([- r, 0]; H_{0}^{1} (Ω))} (0, R_{2} (t)) \end{matrix}$ $\begin{array}{} \displaystyle B_2(t) = \overline{B}_{C([-r,0]; H^1_0(\Omega))}(0, R_2(t)) \end{array}$ is pullback 𝓓-absorbing for the mapping U(t, τ).

The proof of the proposition is completed. ◼

3.5

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 τ_n →_n→+∞ – ∞ and (u^{0, n}, φⁿ ∈ D(τ_n), be fixed. We have to check that the sequence

$\begin{matrix} {S (t, τ_{n}) (u^{0, n}, φ^{n})} = {(u (t, τ_{n}, (u^{0, n}, φ^{n})), u_{t} (., τ_{n}, (u^{0, n}, φ^{n})))}, \end{matrix}$ $$\begin{array}{} \displaystyle \{S(t,\tau_n)(u^{0,n},\varphi^n)\}= \{(u(t,\tau_n, (u^{0,n},\varphi^n)), u_t(.,\tau_n, (u^{0,n},\varphi^n)))\}\,, \end{array}$$

is relatively compact in H. In order to show this, we need to prove that the sequence

$\begin{matrix} {U (t, τ_{n}) (u^{0, n}, φ^{n})} = {u_{t} (., τ_{n}, (u^{0, n}, φ^{n}))} \end{matrix}$ $$\begin{array}{} \displaystyle \{U(t,\tau_n)(u^{0,n},\varphi^n)\} = \{u_t(.,\tau_n, (u^{0,n},\varphi^n))\} \end{array}$$

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 $\begin{matrix} {u_{t} (., τ_{n}, (u^{0, n}, φ^{n}))} := {u_{t}^{n} (.)}, i . e . \forall ε > 0, \exists δ > 0 \end{matrix}$ $\begin{array}{} \displaystyle \{u_t(.,\tau_n, (u^{0,n},\varphi^n))\}:= \{u_t^n(.)\}\,, \rm i.e.\,\, \forall \varepsilon \gt 0, \, \exists \delta \gt 0 \end{array}$ such that if $\begin{matrix} | θ_{1} - θ_{2} | \leq δ, t h e n ∥ u_{t}^{n} (θ_{1}) - u_{t}^{n} (θ_{2}) ∥ \leq ε, f o r a l l θ_{1} > θ_{2} \in [- r, 0] \end{matrix}$ $\begin{array}{} \displaystyle \vert \theta_1 -\theta_2 \vert \leq \delta \,,\, \rm then\, \Vert u^n_t(\theta_1) - u^n_t(\theta_2) \Vert \leq \varepsilon\,,\, for\, all\, \,\theta_1 \gt \theta_2 \in [-r,0]\, \end{array}$;

the uniform boundedness of $\begin{matrix} {u_{t}^{n} (θ)}, \end{matrix}$ $\begin{array}{} \displaystyle \{u^n_t(\theta)\}\,, \end{array}$ 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 $\begin{matrix} {u_{t}^{n}} \end{matrix}$ $\begin{array}{} \displaystyle \{u^n_t\} \end{array}$ and {u_t} are uniformly bounded in C([–r, 0]; L²(Ω)).

To prove (a), we proceed as follows:

$\begin{matrix} ∥ u_{t}^{n} (θ_{1}) - u_{t}^{n} (θ_{2}) ∥ = ∥ u (t + θ_{1}) - u (t + θ_{2}) ∥, \\ = (\int_{t + θ_{2}}^{t + θ_{1}} u^{'} (s) d s), \\ \leq \int_{t + θ_{2}}^{t + θ_{1}} ∥ u^{'} (s) ∥ d s, \\ \leq \int_{t + θ_{2}}^{t + θ_{1}} (∥ Δ u (s) ∥ + ∥ f (u (s)) ∥ + ∥ b (s, u_{s}) ∥ \\ + ∥ g (s) ∥) d s . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u^n_t(\theta_1) - u^n_t(\theta_2) \Vert= \Vert u(t+\theta_1) - u(t+\theta_2)\Vert \,,\nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\,\,\,= \left\Vert \int_{t+\theta_2}^{t+\theta_1} u'(s) ds \right\Vert\,,\nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\,\,\,\leq \int_{t+\theta_2}^{t+\theta_1} \Vert u'(s) \Vert ds \,,\nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\,\,\,\leq \int_{t+\theta_2}^{t+\theta_1} \bigg( \Vert \Delta u(s) \Vert + \Vert f(u(s))\Vert + \Vert b(s,u_s)\Vert \nonumber\\ \displaystyle \qquad\qquad\qquad\quad\,\,\,\,\,+\, \Vert g(s)\Vert\bigg) ds\,. \end{array}$$(3.25)

Now, we estimate the terms on the right hand side of this inequality

From the Holder inequality, we have

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥ d s \leq {(\int_{t + θ_{2}}^{t + θ_{1}} d s)}^{1 / 2} {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥^{2} d s)}^{1 / 2}, \\ \leq | θ_{1} - θ_{2} |^{1 / 2} {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥^{2} d s)}^{1 / 2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert ds \leq \left(\int_{t+\theta_2}^{t+\theta_1} ds\right)^{1/2} \left(\int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert^2 ds\right)^{1/2}\,,\\ \displaystyle \qquad\qquad\qquad\quad\quad\leq \vert \theta_1 - \theta_2 \vert^{1/2} \left(\int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert^2 ds\right)^{1/2}\,. \end{array}$$

On the one hand, we have

$\begin{matrix} ∥ Δ u ∥^{2} \leq λ_{m} ∥ \nabla u ∥^{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert \Delta u \Vert^2 \leq \lambda_m \Vert \nabla u \Vert^2\,. \end{array}$$

So, using this inequality in (3.22) and integrating it over [t + θ₂, t + θ₁], one obtain

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥^{2} d s \leq λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2} d s \\ \leq k_{3} λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} e^{- α^{*} (s - τ)} ∥ \nabla u (τ) ∥^{p} d s + 2 C_{b} η^{- 1} λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} e^{- α (s - τ)} ∥ u (τ) ∥^{2} d s \\ + 2 C_{b} r λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} (e^{- α^{*} (s - τ)} + C_{b} η^{- 1} e^{- α (s - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} d s \\ + 2 c | Ω | λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} ({α^{*}}^{- 1} (1 - e^{- α^{*} (s - τ)}) + C_{b} α^{- 1} η^{- 1} (1 - e^{- α (s - τ)})) d s \\ + 2 λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} e^{- α^{*} s} \int_{- \infty}^{s} e^{α^{*} s^{'}} ∥ g (s^{'}) ∥^{2} d s^{'} d s \\ + 2 C_{b} η^{- 1} λ_{m} \int_{t + θ_{2}}^{t + θ_{1}} e^{- α s} \int_{- \infty}^{s} e^{α s^{'}} ∥ g (s^{'}) ∥^{2} d s^{'} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert^2 ds \leq \lambda_m \int_{t+\theta_2}^{t+\theta_1} \Vert \nabla u(s) \Vert^2 ds \\ \displaystyle \leq k_3 \lambda_m \int_{t+\theta_2}^{t+\theta_1} e^{-\alpha^* (s-\tau)} \Vert \nabla u(\tau) \Vert^p ds + 2C_b \eta^{-1}\lambda_m \int_{t+\theta_2}^{t+\theta_1} e^{-\alpha(s- \tau)} \Vert u(\tau) \Vert^2 ds \\ \displaystyle +\,2C_br\lambda_m \int_{t+\theta_2}^{t+\theta_1} \left(e^{-\alpha^* (s-\tau)} + C_b \eta^{-1} e^{-\alpha (s-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 ds \\ \displaystyle +\, 2c \vert \Omega \vert \lambda_m \int_{t+\theta_2}^{t+\theta_1}\left( {\alpha^*}^{-1} \left ( 1 - e^{-\alpha^* (s-\tau)}\right) + C_b \alpha^{-1} \eta^{-1} \left(1- e^{-\alpha (s-\tau)}\right)\right) ds \\ \displaystyle +\, 2 \lambda_m \int_{t+\theta_2}^{t+\theta_1} e^{-\alpha^* s} \int_{-\infty}^{s} e^{\alpha^* s'}\Vert g(s') \Vert^2 ds'ds \\ \displaystyle +\, 2C_b \eta^{-1} \lambda_m \int_{t+\theta_2}^{t+\theta_1} e^{-\alpha s}\int_{-\infty}^{s} e^{\alpha s'} \Vert g(s') \Vert^2 ds'ds \,. \end{array}$$

Therefore, one obtains

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥^{2} d s \\ \leq k_{3} λ_{m} ∥ \nabla u (τ) ∥^{p} {α^{*}}^{- 1} e^{- α^{*} (t - τ)} (e^{- α^{*} θ_{2}} - e^{- α^{*} θ_{1}}) \\ + 2 C_{b} η^{- 1} λ_{m} ∥ u (τ) ∥^{2} α^{- 1} e^{- α (t - τ)} (e^{- α θ_{2}} - e^{- α θ_{1}}) \\ + 2 C_{b} r λ_{m} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} {α^{*}}^{- 1} e^{- α^{*} (t - τ)} (e^{- α^{*} θ_{2}} - e^{- α^{*} θ_{1}}) \\ + 2 C_{b}^{2} r λ_{m} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} η^{- 1} α^{- 1} e^{- α (t - τ)} (e^{- α θ_{2}} - e^{- α θ_{1}}) \\ + 2 c | Ω | λ_{m} {α^{*}}^{- 1} (1 - {α^{*}}^{- 1} e^{- α^{*} (t - τ)} (e^{- α^{*} θ_{2}} - e^{- α^{*} θ_{1}})) \\ + 2 c | Ω | C_{b} η^{- 1} λ_{m} α^{- 1} (1 - α^{- 1} e^{- α (t - τ)} (e^{- α θ_{2}} - e^{- α θ_{1}})) \\ + 2 λ_{m} {α^{*}}^{- 1} (e^{- α^{*} θ_{2}} - e^{- α^{*} θ_{1}}) e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s^{'}} ∥ g (s^{'}) ∥^{2} d s^{'} \\ + 2 C_{b} η^{- 1} λ_{m} α^{- 1} (e^{- α θ_{2}} - e^{- α θ_{1}}) e^{- α t} \int_{- \infty}^{t} e^{α s^{'}} ∥ g (s^{'}) ∥^{2} d s^{'} \\ \to 0 when θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert^2 ds\\ \displaystyle \leq k_3 \lambda_m \Vert \nabla u(\tau) \Vert^p {\alpha^*}^{-1} e^{-\alpha^*(t- \tau)} \left(e^{-\alpha^* \theta_2} - e^{-\alpha^* \theta_1}\right)\\ \displaystyle +\, 2C_b \eta^{-1}\lambda_m \Vert u(\tau) \Vert^2 \alpha^{-1} e^{-\alpha (t- \tau)} \left(e^{-\alpha \theta_2} - e^{-\alpha \theta_1}\right)\\ \displaystyle +\, 2C_br\lambda_m \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 {\alpha^*}^{-1} e^{-\alpha^*(t- \tau)} \left(e^{-\alpha^* \theta_2} - e^{-\alpha^* \theta_1}\right)\\ \displaystyle +\,2C^2_br\lambda_m \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \eta^{-1} \alpha^{-1} e^{-\alpha (t- \tau)} \left(e^{-\alpha \theta_2} - e^{-\alpha \theta_1}\right)\\ \displaystyle +\, 2c \vert \Omega \vert \lambda_m {\alpha^*}^{-1} \left ( 1 - {\alpha^*}^{-1} e^{-\alpha^*(t- \tau)} \left(e^{-\alpha^* \theta_2} - e^{-\alpha^* \theta_1}\right)\right)\\ \displaystyle +\,2c \vert \Omega \vert C_b \eta^{-1} \lambda_m \alpha^{-1} \left(1- {\alpha}^{-1} e^{-\alpha(t- \tau)} \left(e^{-\alpha \theta_2} - e^{-\alpha \theta_1}\right)\right)\\ \displaystyle +\, 2 \lambda_m {\alpha^*}^{-1} \left( e^{-\alpha^*\theta_2} - e^{-\alpha^*\theta_1} \right) e^{-\alpha^*t} \int_{-\infty}^{t} e^{\alpha^* s'}\Vert g(s') \Vert^2 ds'\\ \displaystyle + 2C_b \eta^{-1} \lambda_m \alpha^{-1} \left( e^{-\alpha \theta_2} - e^{-\alpha \theta_1} \right) e^{-\alpha t} \int_{-\infty}^{t} e^{\alpha s'}\Vert g(s') \Vert^2 ds'\\ \displaystyle \quad\to 0 \; \mbox{when} \; \theta_1 \to \theta_2\,. \end{array}$$(3.26)

Hence, it follows that

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥ d s \leq | θ_{1} - θ_{2} |^{1 / 2} {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ Δ u (s) ∥^{2} d s)}^{1 / 2} \\ \to 0 when θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert ds \leq \vert \theta_1 - \theta_2 \vert^{1/2} \left(\int_{t+\theta_2}^{t+\theta_1} \Vert \Delta u(s) \Vert^2 ds\right)^{1/2}\\ \displaystyle \qquad\qquad\qquad\qquad\quad \to 0 \; \mbox{when} \; \theta_1 \to \theta_2\,. \end{array}$$

From the Holder inequality, we have

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ f (u (s)) ∥ d s \leq | θ_{1} - θ_{2} |^{1 / 2} \cdot {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ f (u (s)) ∥^{2} d s)}^{1 / 2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert f(u(s))\Vert ds \leq \vert \theta_1 - \theta_2 \vert^{1/2} \cdot \left( \int_{t+\theta_2}^{t+\theta_1} \Vert f(u(s))\Vert^2 ds \right)^{1/2} \,. \end{array}$$(3.27)

Using (1.4) and the convexity of the power, one gets

$\begin{matrix} ∥ f (u (t)) ∥^{2} = \int_{Ω} | f (u (t, x)) |^{2} d x \\ \leq 2 l^{2} ∥ u (t) ∥^{2 (p - 1)} + 2 l^{2} | Ω | . \end{matrix}$ $$\begin{array}{} \displaystyle \Vert f(u(t))\Vert^2 = \int_{\Omega} \vert f(u(t,x))\vert^2 dx \,\\ \displaystyle \qquad\qquad\,\,\,\,\leq 2l^2 \Vert u(t) \Vert^{2(p-1)} + 2l^2 \vert \Omega \vert\,. \end{array}$$

Integrating this estimate over [t + θ₂, t + θ₁], one finds

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ f (u (s)) ∥^{2} d s \leq 2 l^{2} \int_{t + θ_{2}}^{t + θ_{1}} ∥ u (s) ∥^{2 (p - 1)} d s + 2 l^{2} | Ω | \cdot | θ_{1} - θ_{2} | . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert f(u(s))\Vert^2 ds \leq 2l^2 \int_{t+\theta_2}^{t+\theta_1} \Vert u(s) \Vert^{2(p-1)} ds + 2l^2 \vert \Omega \vert \cdot \vert \theta_1 -\theta_2 \vert \,. \end{array}$$

Since λ₁∥u∥² ≤ ∥∇u∥², we have

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ f (u (s)) ∥^{2} d s \leq 2 l^{2} λ_{1}^{(p - 1)} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2 (p - 1)} d s + 2 l^{2} | Ω | \cdot | θ_{1} - θ_{2} | . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert f(u(s))\Vert^2 ds \leq 2l^2 \lambda_1^{(p-1)}\int_{t+\theta_2}^{t+\theta_1} \Vert \nabla u(s) \Vert^{2(p-1)} ds + 2l^2 \vert \Omega \vert \cdot \vert \theta_1 -\theta_2 \vert \,. \end{array}$$(3.28)

From (3.22), one has

$\begin{matrix} ∥ \nabla u (t) ∥^{2 (p - 1)} \leq {k_{3} e^{- α^{*} (t - τ)} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} η^{- 1} e^{- α (t - τ)} ∥ u (τ) ∥^{2} \\ + 2 C_{b} r (e^{- α^{*} (t - τ)} + C_{b} η^{- 1} e^{- α (t - τ)}) ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2} \\ + 2 c | Ω | {α^{*}}^{- 1} + 4 C_{b} c | Ω | α^{- 1} η^{- 1} \\ + 2 e^{- α^{*} t} \int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s + 2 C_{b} η^{- 1} e^{- α t} \int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s}^{(p - 1)} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert \nabla u(t) \Vert^{2(p-1)} \leq \bigg\{ k_3 e^{-\alpha^* (t-\tau)} \Vert \nabla u(\tau) \Vert^p + 2C_b \eta^{-1} e^{-\alpha(t- \tau)}\Vert u(\tau) \Vert^2 \\ \displaystyle +\, 2C_br \left(e^{-\alpha^* (t-\tau)} + C_b \eta^{-1} e^{-\alpha (t-\tau)} \right) \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \\ \displaystyle +\, 2c \vert \Omega \vert {\alpha^*}^{-1} + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1} \\ \displaystyle +\, 2 e^{-\alpha^* t} \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds + 2C_b \eta^{-1} e^{-\alpha t}\int_{-\infty}^{t} e^{\alpha s} \Vert g(s) \Vert^2 ds \bigg\}^{(p-1)}\,. \end{array}$$

By applying the convexity of power three times, one gets

$\begin{matrix} ∥ \nabla u (t) ∥^{2 (p - 1)} \\ \leq 2^{2 (p - 2)} {(k_{3} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} r ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2})}^{(p - 1)} e^{- (p - 1) α^{*} (t - τ)} \\ + 2^{2 (p - 2)} {(2 C_{b} η^{- 1} ∥ u (τ) ∥^{2} + 2 C_{b}^{2} r η^{- 1} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2})}^{(p - 1)} e^{- (p - 1) α (t - τ)} \\ + 2^{2 (p - 2)} {(2 c | Ω | {α^{*}}^{- 1} + 4 C_{b} c | Ω | α^{- 1} η^{- 1})}^{(p - 1)} \\ + 2^{3 (p - 2)} 2^{(p - 1)} {(\int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s)}^{(p - 1)} e^{- (p - 1) α^{*} t} \\ + 2^{3 (p - 2)} (2 C_{b} η^{- 1})^{(p - 1)} {(\int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s)}^{(p - 1)} e^{- (p - 1) α t} . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\Vert \nabla u(t) \Vert^{2(p-1)} \\ \displaystyle \leq 2^{2(p-2)}\left(k_3 \Vert \nabla u(\tau) \Vert^p + 2C_br\Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2\right)^{(p-1)}e^{-(p-1)\alpha^* (t-\tau)} \\ \displaystyle +\, 2^{2(p-2)} \left(2C_b \eta^{-1} \Vert u(\tau) \Vert^2 + 2C^2_br \eta^{-1} \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \right)^{(p-1)}e^{-(p-1)\alpha (t-\tau)} \\ \displaystyle +\, 2^{2(p-2)}\left(2c \vert \Omega \vert {\alpha^*}^{-1} + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1}\right)^{(p-1)}\\ \displaystyle + \,2^{3(p-2)} 2^{(p-1)} \left( \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \right)^{(p-1)} e^{-(p-1)\alpha^* t}\\ \displaystyle +\, 2^{3(p-2)} (2C_b \eta^{-1})^{(p-1)} \left( \int_{-\infty}^{t} e^{\alpha s}\Vert g(s) \Vert^2 ds \right)^{(p-1)} e^{-(p-1)\alpha t}\,. \end{array}$$

Integrating it over [t + θ₂, t + θ₁], one obtains

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2 (p - 1)} d s \\ \leq 2^{2 (p - 2)} {(k_{3} ∥ \nabla u (τ) ∥^{p} + 2 C_{b} r ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2})}^{(p - 1)} \int_{t + θ_{2}}^{t + θ_{1}} e^{- (p - 1) α^{*} (s - τ)} d s \\ + 2^{2 (p - 2)} {(2 C_{b} η^{- 1} ∥ u (τ) ∥^{2} + 2 C_{b}^{2} r η^{- 1} ∥ φ ∥_{C ([- r, 0]; L^{2} (Ω))}^{2})}^{(p - 1)} \int_{t + θ_{2}}^{t + θ_{1}} e^{- (p - 1) α (s - τ)} d s \\ + 2^{2 (p - 2)} {(2 c | Ω | {α^{*}}^{- 1} + 4 C_{b} c | Ω | α^{- 1} η^{- 1})}^{(p - 1)} | θ_{1} - θ_{2} | \\ + 2^{3 (p - 2)} 2^{(p - 1)} {(\int_{- \infty}^{t} e^{α^{*} s} ∥ g (s) ∥^{2} d s)}^{(p - 1)} \int_{t + θ_{2}}^{t + θ_{1}} e^{- (p - 1) α^{*} s} d s \\ + 2^{3 (p - 2)} (2 C_{b} η^{- 1})^{(p - 1)} {(\int_{- \infty}^{t} e^{α s} ∥ g (s) ∥^{2} d s)}^{(p - 1)} \int_{t + θ_{2}}^{t + θ_{1}} e^{- (p - 1) α s} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \quad\int_{t+\theta_2}^{t+\theta_1}\Vert \nabla u(s) \Vert^{2(p-1)} ds\\ \displaystyle \leq 2^{2(p-2)}\left(k_3 \Vert \nabla u(\tau) \Vert^p + 2C_br\Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2\right)^{(p-1)} \int_{t+\theta_2}^{t+\theta_1} e^{-(p-1)\alpha^* (s-\tau)} ds \\ \displaystyle +\, 2^{2(p-2)} \left(2C_b \eta^{-1} \Vert u(\tau) \Vert^2 + 2C^2_br \eta^{-1} \Vert \varphi\Vert_{C([-r,0]; L^2(\Omega))}^2 \right)^{(p-1)}\int_{t+\theta_2}^{t+\theta_1} e^{-(p-1)\alpha (s-\tau)} ds \\ \displaystyle +\, 2^{2(p-2)}\left(2c \vert \Omega \vert {\alpha^*}^{-1} + 4C_b c\vert \Omega \vert \alpha^{-1} \eta^{-1}\right)^{(p-1)} \vert \theta_1- \theta_2 \vert \\ \displaystyle +\, 2^{3(p-2)} 2^{(p-1)} \left( \int_{-\infty}^{t} e^{\alpha^* s}\Vert g(s) \Vert^2 ds \right)^{(p-1)} \int_{t+\theta_2}^{t+\theta_1} e^{-(p-1)\alpha^* s} ds\\ \displaystyle +\, 2^{3(p-2)} (2C_b \eta^{-1})^{(p-1)} \left( \int_{-\infty}^{t} e^{\alpha s}\Vert g(s) \Vert^2 ds \right)^{(p-1)} \int_{t+\theta_2}^{t+\theta_1} e^{-(p-1)\alpha s} ds\,. \end{array}$$

Therefore, we get

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2 (p - 1)} d s \leq C_{1}^{'} e^{- (p - 1) α^{*} (t - τ)} (e^{- (p - 1) α^{*} θ_{2}} - e^{- (p - 1) α^{*} θ_{1}}) \\ + C_{2}^{'} e^{- (p - 1) α (t - τ)} (e^{- (p - 1) α θ_{2}} - e^{- (p - 1) α θ_{1}}) \\ + C_{3}^{'} | θ_{1} - θ_{2} | + C_{4}^{'} e^{- (p - 1) α t} (e^{- (p - 1) α θ_{2}} - e^{- (p - 1) α θ_{1}}) \\ + C_{5}^{'} e^{- (p - 1) α^{*} t} (e^{- (p - 1) α^{*} θ_{2}} - e^{- (p - 1) α^{*} θ_{1}}) \\ \to 0 as θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1}\Vert \nabla u(s) \Vert^{2(p-1)} ds \leq C'_1 e^{-(p-1)\alpha^*(t- \tau)} \left(e^{-(p-1)\alpha^* \theta_2} - e^{-(p-1)\alpha^* \theta_1}\right)\\ \displaystyle \qquad\qquad\qquad\qquad\qquad+\,\, C'_2 e^{-(p-1)\alpha(t- \tau)} \left(e^{-(p-1)\alpha \theta_2} - e^{-(p-1)\alpha \theta_1}\right)\\ \displaystyle \qquad\qquad\qquad\qquad\qquad+\,\, C'_3 \vert \theta_1- \theta_2 \vert + C'_4 e^{-(p-1)\alpha t} \left( e^{-(p-1)\alpha \theta_2} - e^{-(p-1)\alpha \theta_1}\right)\\ \displaystyle \qquad\qquad\qquad\qquad\qquad+\,\, C'_5 e^{-(p-1)\alpha^* t} \left( e^{-(p-1)\alpha^* \theta_2} - e^{-(p-1)\alpha^* \theta_1}\right)\\ \displaystyle \qquad\qquad\qquad\qquad\qquad\quad\to 0 \; \mbox{as}\; \theta_1 \to \theta_2\,. \end{array}$$

Hence by (3.27), (3.28) and this last estimate we deduce that

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ f (u (s)) ∥ d s \to 0 as θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert f(u(s))\Vert ds \;\to 0 \; \mbox{as}\; \theta_1 \to \theta_2\,. \end{array}$$

Similarly, by the Holder inequality, we have

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ b (s, u_{s}) ∥ d s \leq | θ_{1} - θ_{2} |^{1 / 2} \cdot {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ b (s, u_{s}) ∥^{2} d s)}^{1 / 2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert b(s,u_s)\Vert ds \leq \vert \theta_1 - \theta_2 \vert^{1/2} \cdot \left( \int_{t+\theta_2}^{t+\theta_1} \Vert b(s,u_s)\Vert^2 ds \right)^{1/2} \,. \end{array}$$(3.29)

On the other hand, by (II), (1.7) and since λ₁∥u∥² ≤ ∥∇u∥², one has

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ b (s, u_{s}) ∥^{2} d s \leq C_{b} \int_{t + θ_{2} - r}^{t + θ_{1}} ∥ u (s) ∥^{2} d s \\ \leq \int_{t + θ_{2} - r}^{t + θ_{2}} ∥ u (s) ∥^{2} d s + \int_{t + θ_{2}}^{t + θ_{1}} ∥ u (s) ∥^{2} d s \\ \leq ∥ φ ∥_{L^{2} ([- r, 0]; L^{2} (Ω))}^{2} + λ_{1}^{- 1} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2} d s . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert b(s,u_s)\Vert^2 ds \leq C_b \int_{t+\theta_2-r}^{t+\theta_1} \Vert u(s) \Vert^2 ds \nonumber\\ \displaystyle \qquad\qquad\qquad\qquad\,\,\,\,\leq \int_{t+\theta_2-r}^{t+\theta_2} \Vert u(s) \Vert^2 ds + \int_{t+\theta_2}^{t+\theta_1} \Vert u(s) \Vert^2 ds \nonumber\\ \displaystyle \qquad\qquad\qquad\qquad\,\,\,\,\leq \Vert \varphi \Vert^2_{L^2([-r,0]; L^2(\Omega))} + \lambda_1^{-1} \int_{t+\theta_2}^{t+\theta_1} \Vert \nabla u(s) \Vert^2 ds\,. \end{array}$$(3.30)

By (3.26), it follows that

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ \nabla u (s) ∥^{2} d s \to 0 as θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert \nabla u(s) \Vert^2 ds \; \to 0 \; \mbox{as}\; \theta_1 \to \theta_2 \,. \end{array}$$

Then, (3.29), (3.30) and this last estimate, we deduce that

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ b (s, u_{s}) ∥ d s \to 0 when θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert b(s,u_s)\Vert ds \; \to 0 \; \mbox{when}\; \theta_1 \to \theta_2 \,. \end{array}$$

Finally, we use the Holder inequality to obtain

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ g (s) ∥ d s \leq | θ_{1} - θ_{2} |^{1 / 2} \cdot {(\int_{t + θ_{2}}^{t + θ_{1}} ∥ g (s) ∥^{2} d s)}^{1 / 2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert g(s)\Vert ds \leq \vert \theta_1 - \theta_2 \vert^{1/2} \cdot \left(\int_{t+\theta_2}^{t+\theta_1} \Vert g(s)\Vert^2 ds\right)^{1/2} \,. \end{array}$$

Since $\begin{matrix} g \in L_{l o c}^{2} (R; L^{2} (Ω)), \end{matrix}$ $\begin{array}{} \displaystyle g\in L^2_{loc}(\mathbb{R}; L^2(\Omega))\,, \end{array}$ one gets

$\begin{matrix} \int_{t + θ_{2}}^{t + θ_{1}} ∥ g (s) ∥ d s \leq | θ_{1} - θ_{2} |^{1 / 2} \cdot ∥ g ∥_{L^{2} ([t + θ_{2}, t + θ_{1}]; L^{2} (Ω))} \\ \to 0 when θ_{1} \to θ_{2} . \end{matrix}$ $$\begin{array}{} \displaystyle \int_{t+\theta_2}^{t+\theta_1} \Vert g(s)\Vert ds \leq \vert \theta_1 - \theta_2 \vert^{1/2} \cdot \Vert g \Vert_{L^2([t+\theta_2,t+\theta_1]; L^2(\Omega))}\\ \displaystyle \qquad\qquad\qquad\quad\,\,\,\,\,\, \to 0 \; \mbox{when}\; \theta_1 \to \theta_2 \,. \end{array}$$

Consequently, by 1), 2), 3), 4) and (3.25), we deduce that

$\begin{matrix} ∥ u (t + θ_{1}) - u (t + θ_{2}) ∥ \to 0 when θ_{1} \to θ_{2}, \end{matrix}$ $$\begin{array}{} \displaystyle \Vert u(t+\theta_1) - u(t+\theta_2)\Vert \; \to 0 \; \mbox{when}\; \theta_1 \to \theta_2 \,, \end{array}$$

and this ensures the equicontinuity property in C([–r, 0]; L²(Ω)); i.e. the sequence {U(t, τ_n)(u^{0, n}, φⁿ)} is relatively compact in C([–r, 0]; L²(Ω)).

Since we have S(t, τ_n)(u^{0, n}, φⁿ) = j(U(t, τ_n)(u^{0, n}, φⁿ)), so {S(t, τ_n)(u^{0, 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, τ_n)(u^{0, 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-attractorÂ = {A(t) : t ∈ ℝ} in H. Furetheremore, Â ⊂ L²(Ω) × C([–r, 0]; L²(Ω)).

eISSN:: 2444-8656
Language:: English

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Attractors for a nonautonomous reaction-diffusion equation with delay

Published Online: Oct 03, 2018

Page range: 127 - 150

Received: Jan 07, 2018

Accepted: May 04, 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00010

Keywordspullback attractors, nonclassical reaction-diffusion equations, critical exponent, delay term

© 2018 H. Harraga and M. Yebdri, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
pullback attractors, nonclassical reaction-diffusion equations, critical exponent, delay term