A Non-Autonomous Wave Equation With Time-Dependent Energy Damping

Abstract

Motivated by the class of energy models proposed by Balakrishnan-Taylor (1989), Bass and Zes (1991), and Krasovskii (1963), in this work we study the well-posedness and existence of pullback attractor for the wave equation with time-dependent energy damping. The theory of pullback dynamics is based on the recent results established by Carvalho et al. (2013). More specifically, we prove that the evolution process associated with the proposed problem is strongly pullback bounded dissipative and pullback asymptotically compact. This is the first work on pullback dynamics for an energy-damped wave equation.

Keywords

energy damping pullback attractor pullback dynamics wave equation well-posedness

1. Introduction

Problems involving evolution models with structural damping defined by the energy of the stationary model associated have been addressed by several researchers in recent years, see e.g. Balakrishnan (1988), Balakrishnan and Taylor (1989), Bass and Zes (1991), Chueshov (2015), Krasovskii (1963), Sun and Yang (2022a), and Tang et al. (2023). It is well known that damping in evolutionary models affects long-term behavior, specifically fatigue. When damping terms are nonlocal we have a major realism of the model in question, since relevant models in Continuum Mechanics, Mathematical Physics, and Biology are spatially nonlocal, see e.g. Fernández-Cara et al. (2016), Gilboa and Osher (2008) and Lu (2005). In this article, we consider a non-autonomous semilinear model associated with wave propagation energy damping under Dirichlet boundary conditions. Sufficient conditions for strongly pullback bounded dissipative and pullback asymptotically compact of the nonlinear evolution process associated with our model are obtained.

1.1. The Model

Carvalho et al. (2013, Chapter 15) considered the following non-autonomous damped wave equation for a scalar variable $u$ posed on a smooth bounded domain $Ω \subset R^{3}$ :

u_{t t} - Δ u + β (t) u_{t} = f (u), u |_{t = 0} = u_{0}

(1.1)

subject to the boundary condition

u |_{Γ} = 0,

(1.2)

and smooth dynamics properties were studied, in the sense of pullback attractors.

In this work, our main objective is to consider model (1.1) to (1.2) in the context of energy damping models. More specifically, let $Ω \subset R^{n}$ be a domain with smooth boundary $Γ = \partial Ω$ , $n \geq 3$ , we consider the following non-autonomous wave model with nonlocal energy damping

u_{t t} - Δ u + γ (t) E (u, u_{t})^{q} u_{t} + f (u) = 0 in Ω \times [τ, + \infty),

(1.3)

where

q \geq 1 / 2

f (u)

is a nonlinear source term,

γ \in C^{1} (R)

E (u, u_{t}) = (‖ \nabla u ‖^{2} + ‖ u_{t} ‖^{2}) / 2

is the total instantaneous energy associated with the linear part of the system, and

‖ \cdot ‖

represents the norm in

L^{2} (Ω)

. We consider (1.3) with the Dirichlet boundary condition

u = 0 on Γ \times [τ, + \infty),

(1.4)

and the initial conditions

u (x, τ) = u_{τ} (x) and u_{t} (x, τ) = v_{τ} (x), x \in Ω, τ \in R .

(1.5)

Here, we will use the same arguments of Carvalho et al. (2013, Chapter 15) in the analysis of pullback dynamics of boundary-initial value problem (1.3) to (1.5); namely, we consider the pullback attractor theory performed in Carvalho et al. (2013).

In what follows we present the physical justifications for the proposed model and the literature published in connection with this class of problems.

1.2. Physical Motivation

Krasovskii (1963) considered the following class of second-order dissipative equations

\ddot{x} (t) + x (t) + κ (E (t)) \dot{x} (t) = 0,

(1.6)

where the intensity of dissipation is determined by a function

κ

that depends on the energy

E (t)

of the system given by

E (t) = (x^{2} (t) + \dot{x} {(t)}^{2}) / 2

. Energy-type nonlinear damping models of the form

κ (E (t)) \dot{x} (t)

play a central role and their relation to all other nonlinear damping models is thoroughly studied. This class of problems is connected with damping phenomena in flight structures with free response studied by Balakrishnan (1988) and Balakrishnan and Taylor (1989). In Balakrishnan (1988) the proposed model in terms of the basic second-order dynamics for the displacement variable

x (t)

in free response is described by

\ddot{x} (t) + ω^{2} x (t) + γ D (x (t), \dot{x} (t)) = 0,

(1.7)

where

ω

is the mode frequency and

γ > 0

is a (small) damping coefficient

D (x (t), \dot{x} (t))

is a nonlinear damping that can be given by functions like

sign \dot{x} (t), | \dot{x} (t) | \dot{x} (t), | \dot{x} (t) |^{α} \dot{x} (t), α \in (0, 1) .

Later, in Balakrishnan and Taylor (1989) using approximation of Krylov–Bogoliubov, Balakrishnan–Taylor suggested the following class of damping models based on the instantaneous total energy of the system with

D (x (t), \dot{x} (t)) = E {(t)}^{q} \dot{x} (t),

where

q > 0

and

E (t) = (ω^{2} | x (t) |^{2} + | \dot{x} (t) |^{2}) / 2

is the energy associated with system (1.7). Note that, the Krasovskii’s model (1.6) and Balakrishnan–Taylor’s model (1.7) (with

D (x (t), \dot{x} (t)) = E {(t)}^{q} \dot{x} (t)

) are identified in the case

κ (s) \equiv γ s^{q}

1.3. Recent Works

Chueshov (2015, Section 5.3.3) consider the following infinite-dimensional version of the Krasovskii system (1.6)

u_{t t} + A u + κ (‖ u_{t} (t) ‖^{2} + ‖ A^{1 / 2} u (t) ‖^{2}) u_{t} = 0, u |_{t = 0} = u_{0}, u_{t} |_{t = 0} = u_{1},

(1.8)

in a separable Hilbert space

H,

where, as above,

A

is a positive operator with a discrete spectrum,

κ (s)

is a scalar bounded Lipschitz function on

R^{+}

such that

κ (1) = 0

and

κ (s)

is strictly increasing for

s > 1

. On the interval

[0, 1]

the function

κ (s)

can be arbitrary. For instance,

κ (s) = κ_{0} (1 - \frac{2}{1 + s}) or κ (s) = {\begin{array}{lcl} 0, & if & 0 \leq s \leq 1 \\ κ_{0} (1 - \frac{1}{s}), & if & s > 1. \end{array}

The existence of a non-compact global attractor has been established.

In recent years, the presence of energy damping has been considered in the context of autonomous beam/plate models concerning the well-posedness and asymptotic behavior of solutions (see, Bezerra et al., 2023, 2024; Gomes Tavares et al., 2023a, 2023b; Jorge Silva et al., 2019; Sun & Yang, 2022a, 2022b).

More recently, Tang et al. (2023) consider the following infinite-dimensional wave version of the Krasovskii’s model (1.6)

u_{t t} - Δ u + (‖ \nabla u (t) ‖^{p} + ‖ u_{t} (t) ‖^{p}) u_{t} + f (u) = 0,

(1.9)

where

p \in (1, 2)

and

f (u)

is a sourcing term, which is naturally used in the study of the dynamics of problems in the context of attractors. The authors prove the existence of a global attractor and, in addition, develop a new process concerning the dimension near the degenerate point individually and show the infinite dimensionality of the attractor. Note that, in model (1.9), the function

κ

of Krasovskii can be expressed approximately by

κ (E (t)) \approx c E {(t)}^{p / 2}, with p \in (1, 2) .

Motivated by this class of energy models, we propose to study problem (1.9) with the nonlocal dissipation coefficient in its intrinsic form proposed by Balakrishnan–Taylor and multiplied by a time-dependent bounded function $γ (t)$ . The model is established by (1.3)–(1.5).

In the proposed equation (1.3) the function of Krasovskii $κ (s)$ is given by $κ (s) = s^{q}$ , $(q \geq 1 / 2)$ , which is a strictly increasing function for $s \in R^{+}$ with $κ (0) = 0$ . This function extends the class considered in (1.8) and (1.9). Extends the considered in (1.8) because it is not bounded in $R^{+}$ , and extends the considered in (1.9) because contains exponents not covered in (1.9). For example, the case $p = 2$ . Furthermore, the presence of the $γ (t)$ function makes the system non-autonomous, which changes the study context from autonomous to non-autonomous systems. The work is justified as the first analysis of pullback dynamics for this class of energy wave models.

1.4. Plan Work

Our article is organized as follows. In Section 2, we give some preliminary notations, state and prove the well-posedness results for problem (1.3)–(1.5). The existence and uniqueness of mild and regular solutions are obtained through results from the semigroup theory established by Pazy (1983). In Section 3, we first present the basic definitions and results of the theory of pullback dynamics (which we use in this work) established by in Carvalho et al. (2013). Next, we prove the main result of this work, Theorem 3.2, which guarantees that the evolution process ${S (t, τ); t \geq τ \in R}$ generated by the mild solutions of the problem (1.3)–(1.5) has a pullback attractor. The result shows that the process $S (\cdot, \cdot)$ is strongly pullback bounded dissipative and has the pullback asymptotically compact property.

2. Well-Posedness

In this section, we shall formalize assumptions on the nonlinear source term $f (u)$ and the real function $γ (t)$ , and we shall establish well-posedness for the model (1.3)–(1.5). In order, denoting by $λ_{1} > 0$ the first eigenvalue of the operator $- Δ$ with the Dirichlet boundary condition (1.2), then

λ_{1} ‖ u ‖^{2} \leq ‖ \nabla u ‖^{2}, \forall u \in H_{0}^{1} (Ω) .

We will discuss the existence and uniqueness of the solution to problem (1.3)–(1.5) under Assumption 2.1. Below we use the following notations. Let $X$ be a separable Banach space. We denote by $L^{p} (a, b; X)$ with $1 \leq p \leq \infty$ the space of Bochner measurable functions $g : [a, b] \to X$ such that $‖ g (\cdot) ‖_{X} \in L^{p} (a, b)$ . $L^{p} (a, b; X)$ is a Banach space with norm given by

\begin{aligned} ‖ g ‖_{L^{p} (a, b; X)} & = {[\int_{a}^{b} ‖ g (t) ‖_{X}^{p} d t]}^{\frac{1}{p}}, 1 \leq p < \infty, \\ ‖ g ‖_{L^{\infty} (a, b; X)} & = {essup}_{t \in [a, b]} ‖ g (t) ‖_{X} . \end{aligned}

Let also

C (a, b; X)

the space of strongly continuous functions with values in

X

and we define

W^{1, p} (a, b; X) = {g \in C (a, b; X); g^{'} \in L^{p} (a, b; X)} .

Here

g^{'} (t)

is a distributional derivate of

g (t)

with respect to

t

We also denote $W_{0} = L^{2} (Ω)$ , $W_{1} = H_{0}^{1} (Ω)$ , and $W_{2} = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ . Here, the notation $(\cdot, \cdot)$ stands for $L^{2}$ -inner product, and $‖ \cdot ‖_{p}$ denotes $L^{p}$ -norm. Thus, $‖ \nabla \cdot ‖$ and $‖ Δ \cdot ‖$ represent the norms in $W_{1}$ and $W_{2},$ respectively. When there is no possibility of confusion we shall use the same notation $(\cdot, \cdot)$ to represent the duality pairing between any Banach space $W$ and its dual $W^{'}$ . We also consider the following Hilbert spaces

\begin{aligned} H & = W_{1} \times W_{0}, | | (u, v) | |_{H}^{2} = ‖ \nabla u ‖^{2} + ‖ v ‖^{2}, (u, v) \in H, \\ H_{1} & = W_{2} \times W_{1}, | | (u, v) | |_{H_{1}}^{2} = ‖ Δ u ‖^{2} + ‖ \nabla v ‖^{2}, (u, v) \in H_{1} . \end{aligned}

Assumption 2.1
We assume that $γ : R \to R$ is bounded, globally Lipschitz, and
$γ_{0} \leq γ (t) \leq γ_{1} for some γ_{1} \geq γ_{0} > 0.$
(2.1)
We take $f : R \to R$ to be continuously differentiable function with $f (0) = 0$ and polynomial growth
$| f^{'} (u) | \leq C_{f^{'}} (1 + | u |^{ρ}), u \in R,$
(2.2)
for some $C_{f^{'}} > 0$ , and
$\begin{aligned} - C_{f} - \frac{c_{f}}{2} | u |^{2} \leq \hat{f} (u) \leq f (u) u + \frac{c_{f}}{2} | u |^{2}, u \in R, \end{aligned}$
(2.3)
for some constants $C_{f} > 0$ , $c_{f} \in [0, λ_{1})$ and $ρ \leq 2 / (n - 2)$ for $n \geq 3$ .

We set $v = u_{t}$ , define $U = (u, v)$ , then we can rewrite the non-autonomous problem (1.3)–(1.5) as
$\begin{aligned} {\begin{cases} U_{t} = A U + B (t, U), t \geq τ, \\ U (τ) = (u_{τ}, v_{τ}) := U_{τ}, \end{cases} \end{aligned}$
(2.4)
where $A : D (A) \subset H \to H$ is an unbounded linear operator defined by
$A U = {(\begin{matrix} v \\ Δ u \end{matrix})}^{⊥}, U \in D (A) = H_{1},$
(2.5)
and $B : R \times H \to H$ is the nonlinear operator
$B (t, U) = {(\begin{matrix} 0 \\ - γ (t) E {(U)}^{q} v - f (u) \end{matrix})}^{⊥}, t \in R, U = (u, v) \in H .$
(2.6)

We will use the following definition of solutions.
Definition 2.1

A function $U : [τ, T] \to H$ is a regular (classical) solution of (2.4) on $[τ, T)$ if $U$ is continuous on $[τ, T)$ , continuously differentiable on $[τ, T)$ , $U (t) \in D (A)$ for $τ < t < T$ and (2.4) is satisfied on $[τ, T)$ .

Let $A$ be the infinitesimal generator of a $C_{0}$ -semigroup $S (t)$ . Let $U_{τ} \in H$ and $B : [τ, T] \times H \to H$ be continuous in $t$ on $[τ, T]$ and Lipschitz continuous on $H$ . The function $U \in C ([τ, T] : H)$ given by
$U (t) = S (t - τ) U_{τ} + \int_{τ}^{t} S (t - s) B (s, U (s)) d s, τ \leq t \leq T$
is the mild solution of the initial value problem (2.4) on $[τ, T]$ .

Hence, the well-posedness result for (2.4), and consequently for the equivalent non-autonomous system (1.3)–(1.5), reads as follows:
Theorem 2.1
[Well-Posedness] Under Assumption 2.1 we have: (I)
if $U_{τ} \in H$ , then there exists $T_{max} > τ$ such that the non-autonomous problem (2.4) has a unique mild solution $U \in C ([τ, T_{max}), H)$ , which is given by
$U (t) = e^{A (t - τ)} U_{τ} + \int_{τ}^{t} e^{A (t - s)} B (s, U (s)) d s, t \in [τ, T_{max}) .$
(2.7)
Moreover, if $T_{max} < \infty$ then
$lim_{t ↑ T_{max}} | | U (t) | | = \infty .$

(II)
If $U_{τ} \in D (A)$ , then the above mild solution $U$ is a regular solution.

Proof.
The proof of item (I) will be established in three steps below.

Step 1 It is easy to see that operator $A$ is an infinitesimal generator of a $C_{0}$ -group of unitary operators on $H$ , see e.g. Pazy (1983, Theorem 10.8).

Step 2 The nonlinear operator $B (t, \cdot) : H \to H$ given in (2.6) is locally Lipschitz independent of $t \in R$ . Indeed, let us take $R > 0$ and $U^{1} = (u^{1}, u_{t}^{1}), U^{2} = (u^{2}, u_{t}^{2})$ such that $| | U^{1} | |_{H}, | | U^{2} | |_{H} \leq R$ . Denoting $z = u^{1} - u^{2}$ , we have
$| | B (t, U^{1}) - B (t, U^{2}) | |_{H} = ‖ γ (t) [E (U^{1})^{q} u_{t}^{1} - E (U^{2})^{q} u_{t}^{2}] + [f (u^{1}) - f (u^{2})] ‖ .$
(2.8)
Now, let’s estimate the terms on the right-hand side of the above equality. First, note that we can rewrite
$\begin{aligned} γ (t) \int_{Ω} [E (U^{1})^{q} u_{t}^{1} - E (U^{2})^{q} u_{t}^{2}] φ d x \\ = γ (t) E (U^{1})^{q} \int_{Ω} z_{t} φ d x + γ (t) [E (U^{1})^{q} - E (U^{2})^{q}] \int_{Ω} u_{t}^{2} φ d x . \end{aligned}$
Using that $γ (t) \leq γ_{1}$ , it is easy to see that
$\begin{aligned} γ (t) E (U^{1})^{q} \int_{Ω} z_{t} φ d x & \leq \frac{γ_{1}}{2^{q}} | | U^{1} | |_{H}^{2 q} ‖ z_{t} ‖ ‖ φ ‖ \\ \leq \frac{γ_{1} R^{2 q}}{2^{q}} | | U^{1} - U^{2} | |_{H} ‖ φ ‖, \forall φ \in W_{0} . \end{aligned}$
Next, again using that $γ (t) \leq γ_{1}$ and $q \geq 1 / 2$ , we have
$\begin{aligned} γ (t) [E (U^{1})^{q} - E (U^{2})^{q}] \int_{Ω} u_{t}^{2} φ d x \\ = \frac{γ (t)}{2^{q}} [| | U^{1} | |_{H}^{2 q} - | | U^{2} | |_{H}^{2 q}] \int_{Ω} u_{t}^{2} φ d x \\ = \frac{q γ (t)}{2^{q - 1}} \int_{0}^{1} {[θ | | U^{1} | |_{H} + (1 - θ) | | U^{1} | |_{H}]}^{2 q - 1} d θ [| | U^{1} | |_{H} - | | U^{2} | |_{H}] \int_{Ω} u_{t}^{2} φ d x \\ \leq 2^{q + 1} q γ_{1} R^{2 q} | | U^{1} - U^{2} | |_{H} ‖ φ ‖, \forall φ \in W_{0} . \end{aligned}$
Finally, from Mean Value Theorem (MVT), Assumption (2.2), Hölder’s inequality with $ρ / (2 (ρ + 1)) + 1 / (2 (ρ + 1)) + (1 / 2) = 1$ , and embedding $W_{1} ↪ L^{2 (ρ + 1)} (Ω)$ , we get
$\begin{aligned} \int_{Ω} (f (u^{1}) - f (u^{2})) φ d x & \leq C_{f^{'}} \int_{Ω} [1 + {(| u^{1} | + | u^{2} |)}^{ρ}] | z | | φ | d x \\ \leq C {[| Ω | + ‖ u^{1} ‖_{2 (ρ + 1)}^{2 (ρ + 1)} + ‖ u^{2} ‖_{2 (ρ + 1)}^{2 (ρ + 1)}]}^{\frac{ρ}{2 (ρ + 1)}} ‖ z ‖_{2 (ρ + 1)} ‖ φ ‖ \\ \leq C_{R} | | U^{1} - U^{2} | |_{H} ‖ φ ‖, \forall φ \in W_{0} . \end{aligned}$
Thus, returning to (2.8), there exists a constant $C_{R} > 0$ independent of $t \in R$ such that
$| | B (t, U^{1}) - B (t, U^{2}) | |_{H} \leq C_{R} | | U^{1} - U^{2} | |_{H} .$
(2.9)
From Pazy (1983, Chapter 6, Theorem 1.4) the initial value problem (2.4) has a unique mild solution $U$ on $[τ, T_{max})$ .

Step 3 The mild solution is globally defined, that is, $T_{max} = + \infty .$ Indeed, in order we define
$ω := 1 - \frac{c_{f}}{λ_{1}} > 0.$
Multiplying the equation (1.3) by $u_{t}$ and integrating over $Ω$ we get
$\frac{d}{d t} E_{f} (t) = - γ (t) E (U (t))^{q} ‖ u_{t} (t) ‖^{2} \leq 0,$
(2.10)
where $E_{f} (t) := E_{f} (u (t), u_{t} (t))$ is the energy functional associated with (1.3)–(1.5) given by
$E_{f} (t) = E (t) + \int_{Ω} \hat{f} (u) d x,$
(2.11)
where $\hat{f} (u) = \int_{0}^{u} f (τ) d τ$ , which implies that
$E_{f} (t) \leq E_{f} (τ), \forall t \in [τ, T_{max}) .$
(2.12)
Next, from Assumption (2.3) and using that $W_{1} ↪ W_{0}$ , we have
$\int_{Ω} \hat{f} (u) d x \geq - \frac{c_{f}}{2} ‖ u (t) ‖^{2} - C_{f} | Ω | \geq - \frac{c_{f}}{2 λ_{1}} ‖ \nabla u (t) ‖^{2} - C_{f} | Ω | .$
(2.13)
Then, from definition of $E_{f} (t)$ and (2.13), we obtain
$E_{f} (t) \geq \frac{1}{2} ‖ u_{t} (t) ‖^{2} + \frac{ω}{2} ‖ \nabla u (t) ‖^{2} - C_{f} | Ω | \geq \frac{ω}{2} | | U (t) | |_{H}^{2} - C_{f} | Ω | .$
(2.14)
Thus, from (2.14) and (2.12), we obtain
$| | U (t) | |_{H}^{2} \leq \frac{2}{ω} E_{f} (τ) + \frac{2 C_{f} | Ω |}{ω}, \forall t \in [τ, T_{max}) .$
(2.15)
Estimate (2.15) implies that any mild solution is globally bounded in time. Therefore, from Pazy (1983, Chapter 6, Theorem 1.4) we conclude that $T_{max} = + \infty .$ Which completes the proof of $(I)$ .

The regularity of the mild solution for $U_{τ} \in D (A)$ is established in Step 4 below.

Step 4 Let $U_{τ} \in D (A)$ . Deriving the equation (1.3) with respect to $t$ and taking the multiplier $u_{t t} (t)$ in the resulting expression, it holds that
$\frac{1}{2} \frac{d}{d t} | | U_{t} (t) | |_{H}^{2} + γ (t) E (U (t))^{q} ‖ u_{t t} (t) ‖^{2} = I_{1} + I_{2} + I_{3},$
(2.16)
where
$\begin{aligned} I_{1} & = - γ (t) \frac{d}{d t} E (U (t))^{q} \int_{Ω} u_{t} u_{t t} d x, \\ I_{1} & = - γ^{'} (t) E (U (t))^{q} \int_{Ω} u_{t} u_{t t} d x, \\ I_{3} & = - \int_{Ω} f^{'} (u) u_{t} u_{t t} d x . \end{aligned}$
Now let us estimate the terms on the right-hand side of equality (2.16). Firstly, using (2.10), definition of $E_{f} (U^{m} (t))$ , Assumption (2.2), Hölder inequality with $ρ / (2 (ρ + 1)) + 1 / (2 (ρ + 1)) + 1 / 2$ , embedding $W_{1} ↪ L^{2 (ρ + 1)} (Ω)$ , and (2.15), we obtain that
$\begin{aligned} \frac{d}{d t} E (U (t)) + γ (t) E (U (t))^{q} ‖ u_{t} (t) ‖^{2} & = - \int_{Ω} f (u) u_{t} d x \\ = - \int_{Ω} \int_{0}^{1} f^{'} (θ u) d θ u u_{t} d x \\ \leq C_{f^{'}} \int_{Ω} [1 + | u |^{ρ}] | u | | u_{t} | d x \\ \leq C [1 + ‖ u (t) ‖_{2 (ρ + 1)}^{ρ}] ‖ u (t) ‖_{2 (ρ + 1)} ‖ u_{t} (t) ‖ \\ \leq C [\frac{1}{2} ‖ \nabla u (t) ‖^{2} + \frac{1}{2} ‖ u_{t} (t) ‖^{2}] \\ \leq C E (t) . \end{aligned}$

Hence, from estimate (2.15), we have
$\begin{aligned} \frac{d}{d t} E (U (t))^{q} & = q E (U (t))^{q - 1} \frac{d}{d t} E (U (t)) \\ \leq C E (U (t))^{q} \\ \leq C . \end{aligned}$

So, from Assumption (2.1) and embedding $W_{1} ↪ W_{0}$ the term $I_{1}$ can be estimated as follows
$I_{1} \leq γ_{1} C | | U_{t} (t) | |_{H}^{2} .$
Using that $γ^{'} (t) \leq K$ , estimate (2.15), and embedding $W_{1} ↪ W_{0}$ , it is easy to see that
$I_{2} \leq C | | U_{t} (t) | |_{H}^{2} .$
Finally, applying (2.2), Hölder’s inequality with $(ρ / (2 (ρ + 1))) + (1 / (2 (ρ + 1))) + (1 / 2) = 1$ , the embedding $W_{1} ↪ L^{2 (ρ + 1)} (Ω)$ , estimate (2.15), and Young’s inequality, we have
$\begin{aligned} I_{3} & \leq C_{f^{'}} \int_{Ω} [1 + | u |^{ρ}] | u_{t} | | u_{t t} | d x \\ \leq C [1 + ‖ u (t) ‖_{2 (ρ + 1)}^{ρ}] ‖ u_{t} (t) ‖_{2 (ρ + 1)} ‖ u_{t t} (t) ‖_{2} \\ \leq C [1 + ‖ \nabla u (t) ‖_{2}^{ρ}] ‖ \nabla u_{t} (t) ‖ ‖ u_{t t} (t) ‖ \\ \leq C | | U_{t} (t) | |_{H}^{2} . \end{aligned}$
Thus, substituting $I_{1}, I_{2}$ , and $I_{3}$ in (2.16), we obtain
$\frac{d}{d t} | | U_{t} (t) | |_{H}^{2} + γ (t) E (U (t))^{q} ‖ u_{t t} (t) ‖^{2} \leq C | | U_{t} (t) | |_{H}^{2},$
(2.17)
for every $t \in [τ, T]$ . Then, integrating (2.17) on $(τ, t)$ and using Gronwall’s inequality, we arrive at
$| | U_{t} (t) | |_{H}^{2} \leq e^{C (t - τ)} | | U_{t} (τ) | |_{H}^{2} .$
(2.18)
This implies that $U (t)$ is differentiable on $[τ, T]$ . Since $U_{t} \in C ([τ, T], H)$ , $U$ is continuosly differentiable on $[τ, T]$ . Thus, using the continuous differentiability of $U$ and the assumption on the differentiable of $f$ , it follows that $(s, U) \in [τ, T] \times H \mapsto B (s, U (s)) \in H$ is continuously differentiable on $[τ, T] \times H .$ Therefore, from Pazy (1983, Chapter 6, Theorem 1.5) we conclude that the mild solution of (2.4) with $U_{τ} \in D (A)$ is a regular solution of (2.4). This proves (II) and completes the proof of Theorem 2.1.
2.1. Nonlinear Evolution Process

Consequently, from Theorem 2.1 we may define a nonlinear evolution process ${S (t, τ); t \geq τ \in R}$ in $H$ by

S (t, τ) U_{τ} = (u (t), u_{t} (t)) = U (t),

for any

t \geq τ

. This also implies that each bounded subset of

H

has orbit and pullback orbit bounded.

We recall (see Carvalho et al., 2013, p.3) that a nonlinear evolution process $S (\cdot, \cdot)$ on a metric space $X$ is a family ${S (t, τ); t \geq τ \in R}$ of two-parameter continuous maps $S (t, τ) : X \to X$ satisfying

S (τ, τ) x = x, S (t, τ) = S (t, r) S (r, τ), \forall τ \leq r \leq t .

The process

S (\cdot, \cdot)

is said to be closed if for any pair

τ \leq t

and any sequence

x_{n} \to x

X

, such that

S (t, τ) x_{n} \to y

, we have

S (t, τ) x = y

. Moreover,

S (\cdot, \cdot)

is said to be continuous if the mapping

S (t, τ) : X ⇀ X

is continuous for each pair

τ \leq t

fixed.

3. Pullback Attractors

This section is dedicated to proving the existence of a pullback attractor for the nonlinear evolution process ${S (t, τ); t \geq τ \in R}$ generated by the non-autonomous problem (1.3)–(1.5) in $H$ . In order we will present some important abstract results related to the pullback attractor established in Carvalho et al. (2013).

3.1. Abstract Results

Definition 3.1
Carvalho et al. (2013, Definition 1.11) Let $S (\cdot, \cdot)$ be a nonlinear evolution process on a metric space $X$ . Given $t \in R$ , a set $K \subset X$ pullback attracts a set $D$ at time $t$ under $S (\cdot, \cdot)$ if
$lim_{τ \to - \infty} dist (S (t, τ) D, K) = 0.$
(3.19)
$K$ pullback attracts bounded sets at time $t$ if (3.19) holds for each bounded subset $D$ of $X$ . A time-dependent family of subsets of $X$ , $K (\cdot)$ , pullback attracts bounded subsets of $X$ under $S (\cdot, \cdot)$ if $K (t)$ pullback attracts bounded sets at time $t$ under $S (\cdot, \cdot)$ , for each $t \in R$ .
Definition 3.2
Carvalho et al. (2013, Definition 1.12) A family ${A (t); t \in R}$ is the pullback attractor for a nonlinear evolution process $S (\cdot, \cdot)$ if (i)
$A (t)$ is compact for each $t \in R$ ,
(ii)
$A (\cdot)$ is invariant with respect to $S (\cdot, \cdot)$ ,
(iii)
$A (\cdot)$ pullback attracts bounded subsets of $X$ , and
(iv)
$A (\cdot)$ is the minimal family of closed sets with property (iii).

Since these conditions are hard to verify in many cases, there are some equivalent statements.
Definition 3.3
Carvalho and Sonner (2013, Definition 6) Let $S (\cdot, \cdot)$ be a nonlinear evolution process in a metric space $X$ . A family of bounded subsets ${B (t); t \in R}$ is said to be strongly pullback absorbing all bounded subsets of $X$ , if for all bounded $D \subset X$ and $s \leq t$ there exits $T_{D, s} \geq 0$ such that
$S (s, s - r) D \subset B (t) for all r \geq T_{D, s} .$
Processes possessing a family of bounded strongly pullback absorbing subsets are called strongly pullback bounded dissipative.
Definition 3.4
Carvalho et al. (2013, Definition 2.8) A nonlinear evolution process $S (\cdot, \cdot)$ in a metric space $X$ is said to be pullback asymptotically compact if, for each $t \in R$ , each sequence ${s_{k}} \leq t$ with $s_{k} \to - \infty$ as $k \to \infty$ , and each bounded sequence ${x_{k}} \in X$ the sequence ${S (t, s_{k}) x_{k}}$ has a convergent subsequence.

The following theorem gives a sufficient condition for the existence of a compact pullback attractor $A (\cdot)$ that is bounded in the past, i.e.
$⋃_{s \leq t} A (s)$
is bounded for each $t \in R$ .
Theorem 3.1
Carvalho et al. (2013, Theorem 2.23) If a nonlinear evolution process $S (\cdot, \cdot)$ is strongly pullback bounded dissipative and pullback asymptotically compact and $B (\cdot)$ is a family of bounded subsets of $X$ such that, for each $t \in R$ , $B (t)$ pullback attracts bounded subsets of $X$ at time $τ$ for each $τ \leq t$ , then $S (\cdot, \cdot)$ has a compact pullback attractor $A (\cdot)$ such that $A (t) = ω (\bar{B} (t), t)$ and $⋃_{s \leq t} A (s)$ is bounded for each $t \in R$ .

Here $ω (\bar{B} (t), t)$ is the $ω -$ limit set at time $t$ of $\bar{B}$ in $X$ given by
$ω (\bar{B}, t) := ⋂_{σ \leq t} \bar{⋃_{s \leq σ} S (t, s) \bar{B}},$
or, equivalently
$\begin{aligned} ω (\bar{B} (t), t) = & {y \in X : there are sequences {s_{k}} \leq t, s_{k} \to - \infty as k \to \infty, \\ and {x_{k}} in \bar{B}, such that y = lim_{k \to \infty} S (t, s_{k}) x_{k}} . \end{aligned}$

3.2. Existence of a Pullback Attractor

The main result of the present article is the following theorem.

Theorem 3.2 (Pullback attractor)

Under Assumption 2.1 with $2 \leq 2^{q} q < 4$ the nonlinear evolution process ${S (t, τ); t \geq τ \in R}$ generated by the non-autonomous problem (1.3)–(1.5) in $H$ has pullback attractor $A (\cdot)$ in $H$ .

Proof.
The proof is based on two propositions stated below. Proposition 3.3 guarantees that the process $S (\cdot, \cdot)$ is strongly pullback bounded dissipative and Proposition 3.6 shows that $S (\cdot, \cdot)$ has the pullback asymptotically compact property. Then the existence of a pullback attractor follows from Theorem 3.1.
3.2.1. Strong Bounded Dissipativity

Proposition 3.3
Under Assumption 2.1 the evolution process ${S (t, τ) : t \geq τ \in R}$ in $H$ is strongly pullback bounded dissipative.
Proof.
In order, we define the functional
${\tilde{E}}_{f} (t) = E_{f} (t) + C_{f} | Ω | .$
Let $τ \leq t$ . Multiplying the equation (1.3) by $u_{t}$ and integrating over $Ω \times [t, T]$ , we get
${\tilde{E}}_{f} (T) + \int_{t}^{T} γ (s) E {(s)}^{q} ‖ u_{t} (s) ‖^{2} d s = {\tilde{E}}_{f} (t) .$
(3.20)
From Assumption (2.1), we have
$γ (t) E {(t)}^{q} ‖ u_{t} (t) ‖^{2} \geq \frac{γ_{0}}{2^{q}} ‖ u_{t} (t) ‖^{2 (q + 1)} .$
Hence
$\begin{aligned} {\tilde{E}}_{f} (T) + \frac{γ_{0}}{2^{q}} \int_{t}^{T} ‖ u_{t} (s) ‖^{2 (q + 1)} d s \leq {\tilde{E}}_{f} (t) . \end{aligned}$
(3.21)
From Hölder’s inequality with $(q / (q + 1)) + (1 / (q + 1)) = 1$ , we have
$\int_{t}^{T} ‖ u_{t} (s) ‖^{2} d s \leq (T - t)^{\frac{q}{q + 1}} {[\int_{t}^{T} ‖ u_{t} (s) ‖^{2 (q + 1)} d s]}^{\frac{1}{q + 1}} \leq \frac{2^{\frac{q}{q + 1}}}{γ_{0}^{\frac{1}{q + 1}}} (T - t)^{\frac{q}{q + 1}} {\tilde{E}}_{f} {(t)}^{\frac{1}{q + 1}} .$
(3.22)
Integrating (3.21) from $τ$ to $T$ , we get
$(T - τ) {\tilde{E}}_{f} (T) + \frac{γ_{0}}{2^{q}} \int_{τ}^{T} \int_{t}^{T} ‖ u_{t} (s) ‖^{2 (q + 1)} d s d t \leq \int_{τ}^{T} {\tilde{E}}_{f} (t) d t .$
(3.23)
On the other hand, multiplying the equation (1.3) by $u (t)$ and integrating over $[τ, T] \times Ω$ , we have
$\int_{τ}^{T} [‖ \nabla u (s) ‖^{2} + \int_{Ω} f (u (s)) u (s) d x] d s = \int_{τ}^{T} ‖ u_{t} (s) ‖^{2} d s + \sum_{j = 1}^{2} J_{j},$
(3.24)
where
$\begin{aligned} J_{1} & = [\int_{Ω} u (τ) u_{t} (τ) d x - \int_{Ω} u (T) u_{t} (T) d x], \\ J_{2} & = \int_{τ}^{T} γ (s) E {(s)}^{q} \int_{Ω} u_{t} (s) u (s) d x d s . \end{aligned}$
From embedding $W_{1} ↪ W_{0}$ and Assumption (2.3), we have
$\int_{τ}^{T} \int_{Ω} f (u (s)) u (s) d x d s \geq \int_{τ}^{T} \int_{Ω} \hat{f} (u (s)) d x d s - \frac{c_{f}}{2 λ_{1}} \int_{τ}^{T} ‖ \nabla u (s) ‖^{2} d s .$
Returning to (3.24) and using that $κ = 1 - (c_{f} / λ_{1})$ , we get
$\int_{τ}^{T} {\tilde{E}}_{f} (t) d t + \frac{κ}{2} \int_{τ}^{T} ‖ \nabla u (t) ‖^{2} d t \leq C_{f} | Ω | (T - τ) + \frac{3}{2} \int_{τ}^{T} ‖ u_{t} (s) ‖^{2} d s + \sum_{j = 1}^{2} J_{j} .$
(3.25)
To estimate the term $J_{1}$ we use that $W_{1} ↪ W_{0}$ , inequality $κ / 2 ‖ \nabla u ‖^{2} + 1 / 2 ‖ u_{t} ‖^{2} \leq {\tilde{E}}_{f}$ , and that ${\tilde{E}}_{f} (T) \leq {\tilde{E}}_{f} (τ)$ , because ${\tilde{E}}_{f}$ is decreasing like $E_{f}$ . Then, we have
$\begin{aligned} J_{1} & \leq ‖ u (T) ‖ ‖ u_{t} (T) ‖ + ‖ u (τ) ‖ ‖ u_{t} (τ) ‖ \\ \leq \frac{1}{λ_{1}} [‖ \nabla u (T) ‖ ‖ u_{t} (T) ‖ + ‖ \nabla u (τ) ‖ ‖ u_{t} (τ) ‖] \\ \leq \frac{2}{κ^{\frac{1}{2}} λ_{1}} [{\tilde{E}}_{f} (T) + {\tilde{E}}_{f} (τ)] \\ \leq \frac{4}{κ^{\frac{1}{2}} λ_{1}} {\tilde{E}}_{f} (τ) . \end{aligned}$
Now, using that $E {(t)}^{q} \leq K_{B}$ , embedding $W_{1} ↪ W_{0}$ , Assumption (2.1), and Young’s inequality, we have
$\begin{aligned} J_{2} & \leq γ_{1} K_{B} \int_{τ}^{T} ‖ u_{t} (t) ‖ ‖ \nabla u (t) ‖ d t \\ \leq C_{B} \int_{τ}^{T} ‖ u_{t} (t) ‖^{2} d t + \frac{κ}{2} \int_{τ}^{T} ‖ \nabla u (t) ‖^{2} d t . \end{aligned}$

Substituting $J_{1}$ and $J_{2}$ in (3.25), we obtain
$\int_{τ}^{T} {\tilde{E}}_{f} (t) d t \leq C_{f} | Ω | (T - τ) + \frac{4}{κ^{\frac{1}{2}} λ_{1}} {\tilde{E}}_{f} (τ) + C_{B} \int_{τ}^{T} ‖ u_{t} (t) ‖^{2} d t .$
(3.26)
Inserting (3.26) into (3.23) and using (3.22) with $t = τ$ , we obtain that
$\begin{aligned} (T - τ) {\tilde{E}}_{f} (T) + \frac{γ_{0}}{2^{q}} \int_{τ}^{T} \int_{t}^{T} ‖ u_{t} (s) ‖^{2 (q + 1)} d s d t \\ \leq C_{f} | Ω | (T - τ) + \frac{4}{κ^{\frac{1}{2}} λ_{1}} {\tilde{E}}_{f} (τ) + \frac{2^{\frac{q}{q + 1}} C_{B}}{γ_{0}^{\frac{1}{q + 1}}} (T - τ)^{\frac{q}{q + 1}} {\tilde{E}}_{f} (τ)^{\frac{1}{q + 1}} . \end{aligned}$
(3.27)
From Young’s inequality with $(q / (q + 1)) + (1 / (q + 1)) = 1$ , we get
$\frac{\frac{2^{q}}{q + 1} C_{B}}{γ_{0}^{\frac{1}{q + 1}}} (T - τ)^{\frac{q}{q + 1}} {\tilde{E}}_{f} (τ)^{\frac{1}{q + 1}} \leq (T - τ) + \frac{2^{q} C_{B}^{q + 1}}{γ_{0}} {\tilde{E}}_{f} (τ) .$
Substituting in (3.27), we have
${\tilde{E}}_{f} (T) \leq (1 + C_{f} | Ω |) + \frac{K_{B}}{T - τ} {\tilde{E}}_{f} (τ),$
(3.28)
where $K_{B} = (4 / (κ^{1 / 2} λ_{1})) + ((2^{q} C_{B}^{q + 1}) / γ_{0})$ . So taking $R := (1 + C_{f} | Ω |)^{1 / 2}$ there exists $t_{0} (B) > 0$ such that
$| | S (t, τ) U_{τ} | |_{H}^{2} \leq R, for all t - τ \geq t_{0} (B) .$
(3.29)
Therefore, the closed ball
$B_{0} = B_{H} (0; R) .$
is a constant pullback absorbing family. Which proves that ${S (t, τ); t \geq τ \in R}$ is strongly bounded dissipative in the sense of Definition 3.3.
3.2.2. Pullback Asymptotically Compact

We now present a convenient criterion for verifying the pullback asymptotic compactness of the process $S (\cdot, \cdot)$ (Definition 3.4). The result is a version for non-autonomous systems of the criterion for autonomous systems established by Chueshov and Lasiecka (2008, Proposition 2.10). The adaptation of the criterion for non-autonomous systems can be carried out with minimal changes in a similar way to that performed by Ma et al. (2017, Theorem 3.2) with $X = X_{t}$ and we omit the proof here.

Theorem 3.4
Ma et al. (2017, Theorem 3.2) Let $X$ be a Banach spaces and $S (t, τ) : X \to X$ be an evolution process that possesses a pullback absorbing family ${\hat{B}}_{0} = {B_{0} (τ)}_{τ \in R}$ . Suppose that for any $t \in R$ and $ε > 0$ there exists a time $s_{ε} < t$ and a contractive function $ψ_{ε} : B_{0} (s_{ε}) \times B_{0} (s_{ε}) \to R,$ such that
$| | S (t, s_{ε}) x - S (t, s_{ε}) y | |_{X} \leq ε + ψ_{ε} (x, y) .$
Then the process is pullback asymptotically compact.

We recall that:
Definition 3.5
Let $X$ be a Banach space and $B$ be a bounded subset of $X$ . We call a function $ψ (\cdot, \cdot)$ , defined on $X \times X$ , a contractive function on $B \times B$ if for any sequence ${x_{n}}_{n \in N} \subset B$ , there is a subsequence ${x_{n k}}_{k \in N} \subset {x_{n}}_{n \in N}$ such that
$lim_{k \to \infty} lim_{l \to \infty} ψ (x_{n k}, x_{n l}) = 0.$

The main result of this section is set out below.
Proposition 3.5
Let $B \subset H$ be a bounded set and consider the difference $Z = (z, z_{t}) = (u^{1} - u^{2}, u^{1} - u_{t}^{2})$ of two generalized solutions of the non-autonomous problem (1.3)–(1.5) given by $S (t, τ) U_{τ}^{1} = (u^{1} (t), u_{t}^{1} (t)) = U^{1} (t)$ and $S (t, τ) U_{τ}^{2} = (u^{2} (t), u_{t}^{2} (t)) = U^{2} (t)$ with $U_{τ}^{1} = (u_{τ}^{1}, v_{τ}^{1}), U_{τ}^{2} = (u_{τ}^{2}, v_{τ}^{2}) \in B$ . Under the conditions of the Theorem 2.1 with exponent $q$ such that $2 \leq 2^{q} q < 4$ there exists a constants $C > 0$ and a constant $C_{B, τ, T} > 0$ (depending on $τ \leq T$ ), such that the following inequality holds
$\begin{aligned} E_{z} (T) & \leq \frac{C}{(T - τ)^{\frac{1}{q + 1}}} E_{z} (τ)^{\frac{1}{q + 1}} + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2}^{\frac{1}{q + 1}} \\ + \frac{C}{(T - τ)^{\frac{1}{q + 1}}} {| \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s |}^{\frac{1}{q + 1}} + \frac{1}{(T - τ)} | \int_{τ}^{T} \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s d t |, \end{aligned}$
(3.30)
where
$E_{z} (t) = \frac{1}{2} ‖ z_{t} ‖^{2} + \frac{1}{2} ‖ \nabla z ‖^{2} .$

Proof.
Denoting $z = u^{1} - u^{2}$ and $F (z) = f (u^{1}) - f (u^{2})$ , then the difference $Z = (z, z_{t})$ solves the following problem
$\begin{aligned} {\begin{cases} z_{t t} - Δ z + \frac{γ (t)}{2} [E (U^{1})^{q} + E (U^{2})^{q}] z_{t} + \frac{γ (t)}{2} [E (U^{1})^{q} - E (U^{2})^{q}] [u^{1} + u^{2}] \\ + F (z) = 0, in Ω \times [τ, + \infty) \\ z (τ) = z_{τ}^{1}, z_{t} (τ) = z_{τ}^{2} . \end{cases} \end{aligned}$
(3.31)
Multiplying the equation (3.31) by $z_{t}$ and integrating over $Ω \times [t, T]$ we obtain the following equality
$\begin{aligned} E_{z} (t) & - E_{z} (t) + \frac{1}{2} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s \\ + \frac{1}{2} \int_{t}^{T} γ (s) [E (U^{1})^{q} - E (U^{2})^{q}] [‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}] d s = - \int_{t}^{T} \int_{Ω} F (z) z_{t} d x d s . \end{aligned}$
(3.32)
From Mean Value Theorem there exists $θ \in (0.1)$ such that
$\begin{aligned} E (U^{1})^{q} - E (U^{2})^{q} & = \frac{q}{2} {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} [| | U^{1} | |_{H}^{2} - | | U^{2} | |_{H}^{2}] \\ = \frac{q}{2} {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} {[‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} | |^{2}] - [‖ \nabla u^{1} ‖^{2} - ‖ \nabla u^{2} | |^{2}]} . \end{aligned}$
Returning to (3.32) we get
$\begin{aligned} E_{z} (t) - E_{z} (t) + \frac{1}{2} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s \\ + \frac{q}{4} \int_{t}^{T} γ (s) {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} {[‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}]}^{2} d s \\ = - \underset{Q}{\underset{⏟}{\frac{q}{4} \int_{t}^{T} γ (s) {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} [‖ \nabla u^{1} ‖^{2} - ‖ \nabla u^{2} ‖^{2}] [‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}] d s}} \\ - \int_{t}^{T} \int_{Ω} F (z) z_{t} d x d s . \end{aligned}$
(3.33)
From Young’s inequality and using that $(‖ \nabla u^{i} ‖^{2}) / 2 \leq E (U^{i})$ , $i = 1, 2$ , we have
$\begin{aligned} Q & \leq \frac{q}{4} \int_{t}^{T} γ (s) {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} {[‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}]}^{2} d s \\ + \frac{q}{16} \int_{t}^{T} γ (s) {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} {[‖ \nabla u^{1} ‖ + ‖ \nabla u^{2} ‖]}^{2} ‖ \nabla w ‖ d s \\ \leq \frac{q}{4} \int_{t}^{T} γ (s) {[\frac{θ | | U^{1} | |_{H}^{2} + (1 - θ) | | U^{2} | |_{H}^{2}}{2}]}^{q - 1} {[‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}]}^{2} d s \\ + \frac{2^{q} q}{8} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ \nabla z ‖^{2} d s . \end{aligned}$
Substituting $Q$ in (3.33), we obtain
$\begin{aligned} E_{z} (t) - E_{z} (t) + \frac{1}{2} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s \\ \leq \underset{\tilde{Q}}{\underset{⏟}{\frac{2^{q} q}{8} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ \nabla z ‖^{2} d s}} - \int_{t}^{T} \int_{Ω} F (z) z_{t} d x d s . \end{aligned}$
(3.34)
Now let us give an estimate for the term $\tilde{Q}$ . By multiplying the equation (3.31) by
$\frac{2^{q} q}{8} γ (t) [E (U^{1})^{q} + E (U^{2})^{q}] z$
and integrating over $Ω \times [t, T]$ , we get
$\tilde{Q} = \frac{2^{q} q}{8} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s + \sum_{j = 1}^{6} J_{j},$
(3.35)
where
$\begin{aligned} J_{1} & = \frac{2^{q} q}{8} \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] F (z) z d x d s, \\ J_{2} & = \frac{2^{q} q}{16} \int_{t}^{T} γ {(s)}^{2} {[E (U^{1})^{q} + E (U^{2})^{q}]}^{2} \int_{Ω} z_{t} z d x d s, \\ J_{3} & = \frac{2^{q} q}{16} \int_{t}^{T} γ {(s)}^{2} [E (U^{1})^{2 q} - E (U^{2})^{2 q}] \int_{Ω} [u_{t}^{1} + u_{t}^{2}] z d x d s, \\ J_{4} & = \frac{2^{q} q}{8} \int_{t}^{T} γ^{'} (s) [E (U^{1})^{q} + E (U^{2})^{q}] \int_{Ω} z_{t} z d x d s, \\ J_{5} & = \frac{2^{q} q}{8} \int_{t}^{T} γ (s) \frac{d}{d s} [E (U^{1})^{q} + E (U^{2})^{q}] \cdot \int_{Ω} z_{t} z d x d s, \\ J_{6} & = \frac{2^{q} q}{8} [γ (T) [E (U^{1})^{q} + E (U^{2})^{q}]] \int_{Ω} z_{t} (T) z (T) d x \\ - \frac{2^{q} q}{8} [γ (τ) [E (U^{1})^{q} + E (U^{2})^{q}]] \int_{Ω} z_{t} (τ) z (τ) d x \end{aligned}$
In what follows, we will estimate the terms $J_{1}, \dots, J_{6}$ . We will use $C_{B}$ to denote various constants that depend on $B$ .

Using that $f \in C^{1} (R)$ , MVT, Assumptions (2.1) and (2.2), Hölder’s inequality with $((ρ + 1) / (ρ + 2)) + (2 / (ρ + 2)) = 1$ , and embedding $W_{1} ↪ L^{2 (ρ + 1)} (Ω) ↪ L^{ρ + 2} (Ω)$ , we obtain the following estimate for $J_{1}$
$\begin{aligned} J_{1} & \leq C_{B} \int_{t}^{T} [1 + ‖ u^{1} (s) ‖_{ρ + 2}^{ρ} + ‖ u^{2} (s) ‖^{ρ + 2}] ‖ z (s) ‖_{ρ + 2}^{2} d s \\ \leq C_{B} (T - τ) sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2}^{2} . \end{aligned}$
Using that $γ \in C^{1} (R)$ , from Assumption (2.1) it is easy to see that
$J_{2}, J_{3}, J_{4} \leq C_{B} (T - τ) sup_{τ \leq t \leq T} ‖ z (t) ‖ .$
To estimate the term $J_{5}$ first let us give an estimate for $d / d s [E (U^{1})^{q} + E (U^{2})^{q}]$ . In fact, using that each solution $U^{i}$ is a solution to problem (1.3)–(1.5) the following relation is valid
$\frac{1}{2} \frac{d}{d t} | | U^{i} | |_{H}^{2} + γ (t) E (U^{i})^{q} ‖ u_{t}^{i} ‖^{2} = - \int_{Ω} f (u^{i}) u_{t}^{i} d x .$
(3.36)
From Assumption (2.2), Hölder’s inequality with $ρ / (2 (ρ + 1)) + 1 / (2 (ρ + 1)) + (1 / 2)$ , and embedding $W_{1} ↪ L^{2 (ρ + 1)} (Ω)$ , we get
$\begin{aligned} - \int_{Ω} f (u^{i}) u_{t}^{i} d x \leq C_{B} ‖ \nabla u^{i} ‖ ‖ u_{t}^{i} ‖ \leq C_{B} | | U^{i} | |_{H}^{2} . \end{aligned}$
Substituting this last inequality in (3.36), we have
$\frac{1}{2} \frac{d}{d t} | | U^{i} | |_{H}^{2} \leq C_{B} | | U^{i} | |_{H}^{2} .$
Hence, using that $E (U^{i}) = | | U^{i} | |_{H}^{2} / 2$ , we have
$\frac{d}{d s} [E (U^{1})^{q} + E (U^{2})^{q}] = \frac{q}{2} E (U^{1})^{q - 1} \frac{d}{d t} | | U^{1} | |_{H}^{2} + \frac{q}{2} E (U^{2})^{q - 1} \frac{d}{d t} | | U^{2} | |_{H}^{2} \leq C_{B} .$
Thus, from Assumption (2.1), we can estimate the term $J_{5}$ as follows
$J_{5} \leq C_{B} (T - τ) sup_{τ \leq t \leq T} ‖ z (t) ‖ .$
Finally, using Assumption (2.1) and that $‖ z (T) ‖, ‖ z (τ) ‖ \leq sup_{τ \leq t \leq T} ‖ z (t) ‖$ , we have
$J_{6} \leq C_{B} sup_{τ \leq t \leq T} ‖ z (t) ‖ .$
Substituting the terms $J_{1}, \dots, J_{6}$ in (3.35) and using that $L^{ρ + 2} (Ω) ↪ L^{2} (Ω)$ , we obtain the following estimate for $\tilde{Q}$
$\tilde{Q} \leq \frac{2^{q} q}{8} \int_{t}^{T} [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} .$
Substituting $\tilde{Q}$ in (3.34), we obtain
$\begin{aligned} E_{z} (t) - E_{z} (t) + \frac{1}{2} (1 - \frac{2^{q} q}{4}) \int_{t}^{T} γ (s) [E (U^{1})^{q} + E (U^{2})^{q}] ‖ z_{t} ‖^{2} d s \\ \leq C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} - \int_{t}^{T} \int_{Ω} F (z) z_{t} d x d s . \end{aligned}$
Still, using that $a^{2 q} + b^{2 q} \geq (1 / (2^{2 q - 1})) (a^{2 q} + b^{2 q})$ , we get
$E (U^{1})^{q} + E (U^{2})^{q} \geq \frac{1}{2^{q}} [| | U^{1} | |_{H}^{2 q} + | | U^{2} | |_{H}^{2 q}] \geq \frac{1}{2^{q}} [‖ u_{t}^{1} ‖^{2 q} + ‖ u_{t}^{2} ‖^{2 q}] \geq \frac{1}{2^{3 q - 1}} ‖ z_{t} ‖^{2 q},$
Then, taking $C_{q} = γ_{0} / 2^{3 q} (1 - (2^{q} q / 4))$ , from Assumption (2.1), we obtain
$E_{z} (T) + C_{q} \int_{t}^{T} ‖ z_{t} ‖^{2 (q + 1)} d s \leq E_{z} (t) + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} - \int_{t}^{T} \int_{Ω} F (z) z_{t} d x d s .$
(3.37)
Next, integrating (3.37) from $τ$ to $T$ , we get
$\begin{aligned} (T - τ) E_{z} (T) + C_{q} \int_{τ}^{T} \int_{t}^{T} ‖ z_{t} ‖^{2 (q + 1)} d s d t \\ \leq \int_{τ}^{T} E_{z} (t) d t + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} + | \int_{τ}^{T} \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s d t | . \end{aligned}$
(3.38)
Now, multiplying the equation (3.31) by $z$ and integrating from $τ$ to $T$ , we obtain that
$\int_{τ}^{T} E_{z} (t) d t = 2 \int_{τ}^{T} ‖ z_{t} ‖^{2} d t + \sum_{j = 1}^{4} L_{j},$
(3.39)
where
$\begin{aligned} L_{1} & = \int_{τ}^{T} F (z) z d x d t, \\ L_{2} & = \frac{1}{2} \int_{τ}^{T} γ (t) [E (U^{1})^{q} + E (U^{2})^{q}] z_{t} z d x d t, \\ L_{3} & = \frac{1}{2} \int_{τ}^{T} γ (t) [E (U^{1})^{q} - E (U^{2})^{q}] [u_{t}^{1} + u_{t}^{2}] z d x d t, \\ L_{4} & = \int_{Ω} z_{t} (τ) z (τ) d x - \int_{Ω} z_{t} (T) z (T) d x . \end{aligned}$
In what follows we will estimate the terms on the right-hand side of (3.39). By using Young’s inequality with $(q / (q + 1)) + (1 / (q + 1)) = 1$ and (3.37) with $t = τ$ , we obtain
$\begin{aligned} 2 \int_{τ}^{T} ‖ z_{t} ‖^{2} d s \leq 2 (T - τ)^{\frac{q}{q + 1}} {(\int_{t}^{T} ‖ z_{t} ‖^{2 (q + 1)} d s)}^{\frac{1}{q + 1}} \\ \leq \frac{2 (T - τ)^{\frac{q}{q + 1}}}{C_{q}^{\frac{1}{q + 1}}} {[E_{z} (τ) + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} + | \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s |]}^{\frac{1}{q + 1}} . \end{aligned}$
(3.40)
By arguments similar to those used in estimating the terms $J_{j}^{'} s$ , we have
$\sum_{j = 1}^{4} L_{j} \leq C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} .$
(3.41)
Substituting (3.40) and (3.41) in (3.39), we get
$\begin{aligned} \int_{τ}^{T} E_{z} (t) d t & \leq C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} \\ + \frac{2 (T - τ)^{\frac{q}{q + 1}}}{C_{q}^{\frac{1}{q + 1}}} {[E_{z} (t) + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2} + | \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s |]}^{\frac{1}{q + 1}} . \end{aligned}$
(3.42)
Inserting (3.42) into (3.38) and using that the concave function $h (s) = s^{1 / q + 1}$ satisfies $h (s_{1} + s_{2}) \leq h (s_{1}) + h (s_{2})$ , we have
$\begin{aligned} (T - τ) E_{z} (T) & \leq \frac{2 (T - τ)^{\frac{q}{q + 1}}}{C_{q}^{\frac{1}{q + 1}}} E_{z} (τ)^{\frac{1}{q + 1}} + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2}^{\frac{q}{q + 1}} \\ + \frac{2 (T - τ)^{\frac{q}{q + 1}}}{C_{q}^{\frac{1}{q + 1}}} {| \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s |}^{\frac{1}{q + 1}} + | \int_{τ}^{T} \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s d t | . \end{aligned}$
(3.43)
From (3.43) we still get that
$\begin{aligned} E_{z} (T) & \leq \frac{2}{C_{q}^{\frac{1}{q + 1}} (T - τ)^{\frac{1}{q + 1}}} E_{z} (τ)^{\frac{1}{q + 1}} + C_{B, τ, T} sup_{τ \leq t \leq T} ‖ z (t) ‖_{ρ + 2}^{\frac{q}{q + 1}} \\ + \frac{2}{C_{q}^{\frac{1}{q + 1}} (T - τ)^{\frac{q}{q + 1}}} {| \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s |}^{\frac{1}{q + 1}} + \frac{1}{(T - τ)} | \int_{τ}^{T} \int_{τ}^{T} \int_{Ω} F (z) z_{t} d x d s d t | . \end{aligned}$
Therefore, taking $C = 2 / (C_{q}^{1 / q + 1})$ we obtain the inequality (3.30) and the proof of the proposition is complete.
Proposition 3.6
Under the hypotheses of Proposition 3.5 the corresponding nonlinear evolution process $S (\cdot, \cdot)$ is pullback asymptotically compact in $H$ .
Proof.
Since $Z (τ) = Z_{τ} = U_{τ}^{1} - U_{τ}^{2} \in B$ there exists a constant $K_{B} > 0$ such that $E_{z} (τ) = 1 / 2 | | Z (τ) | |_{H} \leq K_{B}$ . Hence, from (3.30), we have
$\frac{1}{2} | | Z (t) | |_{H} \leq \frac{C K_{B}}{(t - τ)^{\frac{1}{q + 1}}} + Ψ_{t, τ, B} (U_{τ}^{1}, U_{τ}^{2}), τ \leq t .$
(3.44)
where we define
$\begin{aligned} Ψ_{t, τ, B} (U_{τ}^{1}, U_{τ}^{2}) & := C_{B, τ, t} sup_{τ \leq t \leq t} ‖ z (t) ‖_{ρ + 2}^{\frac{q}{q + 1}} + \frac{C}{(t - τ)^{\frac{q}{q + 1}}} {| \int_{τ}^{t} \int_{Ω} F (z) z_{t} d x d s |}^{\frac{1}{q + 1}} \\ + \frac{1}{(t - τ)} | \int_{τ}^{t} \int_{s}^{t} \int_{Ω} F (z) z_{t} d x d r d t |, τ \leq t . \end{aligned}$
From (3.44), given $ε > 0$ there exist $τ_{ε} \leq t$ such that
$| | S (t, τ_{ε}) U_{τ_{ε}}^{1} - S (t, τ_{ε}) U_{τ_{ε}}^{2} | |_{H_{c}} \leq ε + Ψ_{t, τ_{ε}, B} (U_{τ_{ε}}^{1}, U_{τ_{ε}}^{2}) .$
(3.45)

Therefore according to the Theorem 3.1, only if the function $Ψ_{t, τ_{ε}, B} (\cdot, \cdot)$ is a contractive function then the proof of Proposition 3.6 is complete. It remains to show that $Ψ_{t, τ, B}$ is contractive on $B$ . To this end, let $U_{τ_{ε}}^{n} = ({u^{n}}_{τ_{ε}}, {u_{t}^{n}}_{τ_{ε}})$ be a sequence in $B$ . By estimate (3.29) we have
$| | S (s, τ_{ε}) U_{τ_{ε}}^{n} | |_{H}^{2} \leq C_{R}, for all s \in [τ_{ε}, t] .$
(3.46)
It follows that ${(u^{n}, u_{t}^{n})}$ is bounded in $L^{2} (τ_{ε}, t; H)$ . Note that the compact embedding $W_{1} = H_{0}^{1} (Ω) ↪ L^{ρ + 2} (Ω)$ is valid for $1 < ρ < N / (N - 2)$ . Then, by the Aubin–Lions Theorem, there exists a subsequence ${u^{n_{k}}}$ such that converges for some $u$ in $L^{2} (τ_{ε}, t; L^{ρ + 2} (Ω))$ . Thus, we have
$lim_{k \to \infty} lim_{l \to \infty} C_{B, τ_{ϵ}, t} sup_{τ_{ε} \leq s \leq t} ‖ u^{n_{k}} (s) - u^{n l} (s) ‖_{ρ + 2}^{\frac{1}{q + 1}} = 0.$
(3.47)

On the other hand, by using Lemma 8.1 in Lions and Magenes (1972, Page 275), (3.46) also implies that $u^{n}$ is bounded in $C_{τ_{ε}} (τ_{ε}, T; W_{1})$ , and then $u^{n} (t)$ is bounded in $W_{1}$ for all $t \in [τ_{ε}, T] .$ From this one gets
$u^{n} (t) \to u (t) weakly in W_{1}, τ_{ε} \leq t \leq T,$
(3.48)
and due to the compact embedding theorem, we infer
$\hat{f} (u^{n} (t)) \to \hat{f} (u (t)) strongly in L^{1} (Ω), τ_{ε} \leq t \leq T .$
(3.49)
where we remember that $\hat{f} (u) = \int_{0}^{u} f (τ) d τ$ .

Also, from assumptions on $f$ , we have
$(f (u^{n}), u_{t}^{n}) \to (f (u), u_{t}) strongly in L^{1} (τ_{ε}, T) .$
(3.50)
Now, regarding that
$\frac{d}{d t} \int_{Ω} \hat{f} (u^{n} (x, t)) d x = (f (u^{n} (t)), u_{t}^{n} (t)),$
we get
$\int_{τ_{ε}}^{t} (f (u^{n} (τ)), u_{t}^{n} (τ)) d τ = \int_{Ω} \hat{f} (u^{n} (x, t)) d x - \int_{Ω} \hat{f} (u^{n} (τ_{ε}, x)) d x .$
From this identity (which also holds true for $u$ ) and from the limits (3.49)–(3.50), we finally arrive at
$\begin{aligned} lim_{n \to \infty} lim_{m \to \infty} & \int_{τ_{ε}}^{T} (f (u^{n} (t)) - f (u^{m} (t)), u_{t}^{n} (t) - u_{t}^{m} (t)) d t \\ = lim_{n \to \infty} \int_{Ω} \hat{f} (u^{n} (x, T)) d x + lim_{m \to \infty} \int_{Ω} \hat{f} (u^{m} (x, T)) d x \\ - lim_{n \to \infty} \int_{Ω} \hat{f} (u^{n} (x, τ_{ε})) d x - lim_{m \to \infty} \int_{Ω} \hat{f} (u^{m} (x, τ_{ε})) d x \\ - lim_{n \to \infty} lim_{n \to \infty} \int_{τ_{ε}}^{T} \int_{Ω} f (u^{n} (t, x)) u_{t}^{m} (x, t) d x d t \\ - lim_{n \to \infty} lim_{n \to \infty} \int_{τ_{ε}}^{T} \int_{Ω} f (u^{m} (x, t)) u_{t}^{n} (x, t) d x d t \\ = 2 \int_{Ω} f (u (x, T)) d x - 2 \int_{Ω} f (u (x, τ_{ε})) d x \\ - 2 \int_{τ_{ε}}^{T} (f (u (t)), u_{t} (t)) d x d t \\ = 0. \end{aligned}$
(3.51)
Hence, using (3.47) and (3.51), we obtain
$lim inf_{m \to \infty} inf_{n \to \infty} Ψ_{T, τ_{ϵ}, B} (U_{τ_{ε}}^{n}, U_{τ_{ε}}^{m}) = 0.$
Which proves that $Ψ_{T, τ_{ε}, B} (\cdot, \cdot)$ is a contractive function. Therefore, the proof of Proposition 3.6 is now complete.
Remark 3.7 (Further comments)

In this article, we considered the existence of pullback attractor for (2.4) on the space $H$ using arguments from Chueshov and Lasiecka (2008); Lions and Magenes (1972); Ma et al. (2017), differently of the ideas in Carvalho et al. (2013). Inspired by Carvalho et al. (2013, Sections 15.5 and 15.6) it is natural to ask whether the pullback attractor is gradient-like, this will require higher regularity of the attractor; namely, it is possible to prove that $⋃_{t \in R} A (t)$ is a bounded subset of $H_{1}$ ? These issues are still open to study.

Also, it is natural to ask whether the pullback attractor is upper semicontinuous concerning the scalar parameter $q$ or functional parameter $γ$ , this will require higher regularity of the attractor. These issues also are still open to study.

Footnotes

Author Contributions

All authors contributed to the study conception and design.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Vando Narciso was supported by the Fundect/CNPq 15/2024.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data used to support the findings of this study are included in the article.

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