Dynamics for a class of energy beam models with rotational forces

Abstract

This paper is concerned with the well-posedness and long-time dynamics of a class of beam/plate equations with rotational inertia and nonlinear energy damping. The model is derived from nonlocal dissipative energy models for flight structures, as proposed by Balakrishnan-Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). Our main results address the existence of compact global attractors. The work complements the degenerate coefficient case left open by Sun and Yang (J. Math. Anal. Appl., Volume 512, Issue 2, 2022).

Keywords

Energy damping global attractor long-time dynamics plate equation rotational inertia

1. Introduction

1.1. The model

In this work we study well-posedness and long-time dynamics of the following nonlinear beam/plate with nonlocal damping and rotational inertia

\begin{aligned} (I - γ Δ) u_{t t} + Δ^{2} u - κ Δ u - {‖ (u, u_{t}) ‖}_{H_{γ}}^{2 q} Δ u_{t} + f (u) = h, in Ω \times R, \end{aligned}

(1.1)

where

Ω \subset R^{n}

is a bounded domain with smooth boundary

Γ = \partial Ω

γ \in (0, 1]

, κ is a nonnegative constant, q is a positive exponent,

‖ (\cdot, \cdot) ‖_{H_{γ}}

is the norm in weak phase space given by

\begin{aligned} {‖ (u, u_{t}) ‖}_{H_{γ}} = ‖ Δ u ‖^{2} + ‖ u_{t} ‖^{2} + γ ‖ \nabla u_{t} ‖^{2}, \end{aligned}

f (u)

is a nonlinear source, h is an external force, and

‖ \cdot ‖

stands for the norm in

L^{2} (Ω)

. Corresponding to the displacement

u = u (x, t)

we consider the hinged boundary condition, described by

\begin{aligned} u = Δ u = 0, on Γ \times R^{+}, \end{aligned}

(1.2)

and initial condition

\begin{aligned} u (x, 0) = u_{0} (x) and u_{t} (x, 0) = u_{1} (x), x \in Ω . \end{aligned}

(1.3)

We concentrate on the case when $0 < γ ⩽ 1$ . The limiting case $γ \to 0$ (without the presence of rotational forces) and with null external force was recently studied by Jorge Silva et al. [16]

\begin{aligned} u_{t t} - κ Δ u + Δ^{2} u - γ {[\int_{Ω} (| Δ u |^{2} + | u_{t} |^{2}) d x]}^{q} Δ u_{t} + f (u) = 0, \end{aligned}

(1.4)

where the clamped boundary condition was considered. Existence, uniqueness, and polynomial decay rates have been established for regular solutions. The main purpose in the present paper is to study the well-posedness and asymptotic behavior of solutions for (1.1)–(1.3). More specifically, we prove that the dynamical systems

(H_{γ}, S_{t})

generated by the problem (1.1)–(1.3) has a compact global attractor

{A_{γ}}_{γ \in (0, 1]}

. This work complements the recent work of Sun and Yang [18] who consider the case with non degenerate nolocal coefficient.

In current literature, there are many works in the context of beams/plates whose dissipation coefficient is given by a nonlocal function. In this work we will only focus on beam/plate models whose nonlocal damping intensity is determined by the energy of the system.

1.2. Physical motivation

Balakrishnan in [2] studied the damping phenomena in flight structures with free response. The model of Ordinary Differential Equations (ODE) proposed in terms of the basic second-order dynamics for the displacement variable $y (t)$ is described by

\begin{aligned} \ddot{y} (t) + ω^{2} y (t) + γ D (y (t), \dot{y} (t)) = 0, \end{aligned}

(1.5)

where ω is the mode frequency,

γ > 0

is a (small) damping coefficient, and

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

is a nonlinear damping that can be given by functions like

\begin{aligned} sign \dot{y} (t), | \dot{y} (t) | \dot{y} (t), {| \dot{y} (t) |}^{α} \dot{y} (t), α \in (0, 1) . \end{aligned}

Later, Balakrishnan and Taylor in [3], using approximation of Krylov-Bogoliubov, suggested a new class of damping models called energy models based on the instantaneous total energy of the system with

\begin{aligned} D (y (t), \dot{y} (t)) = \dot{y} (t) E {(t)}^{q} = \dot{y} (t) {[\frac{ω^{2} y {(t)}^{2} + \dot{y} {(t)}^{2}}{2}]}^{q}, \end{aligned}

(1.6)

where

q > 0

and

E (t) = \frac{ω^{2} y {(t)}^{2} + \dot{y} {(t)}^{2}}{2}

is the instantaneous energy associated with system (1.5). Here, it is important to mention that the class of ODEs (1.5) with degenerate damping coefficient (1.6) is associated with the following model proposed by Krasovskii [13] (1963)

\begin{aligned} \ddot{y} (t) + y (t) + κ (\dot{y} {(t)}^{2} + y {(t)}^{2}) \dot{y} (t) = 0, \end{aligned}

(1.7)

with damping coefficient given by a function

κ (s)

possessing appropriate conditions. ODE models (1.5) (with D given by (1.6)) and (1.7) are identified in the case

κ (s) = γ s^{q}

In [3], Balakrishnan and Taylor also propose several models for the beam torsion as well as beam bending modes for a uniform Bernoulli beam. In particular, to yield the model (1.5) with $D (y (t), \dot{y} (t)) = \dot{y} (t) E {(t)}^{q}$ for the single mode response, they suggest for beam bending for a beam of length $2 L$ the following model

\begin{aligned} u_{t t} - 2 ξ \sqrt{μ} u_{x x} + μ u_{x x x x} - γ {[μ \int_{- L}^{L} u_{x x}^{2} d x + \int_{- L}^{L} u_{t}^{2} d x]}^{q} u_{x x t} = 0, \end{aligned}

(1.8)

where

u = u (x, t)

represents the transversal deflection of a beam in the rest position,

γ > 0

is a damping coefficient, ζ is a constant appearing in the approximation of Krylov-Bogoliubov and

μ = \frac{2 ζ w}{σ^{2}}

with w being the mode frequency and

σ^{2}

the spectral density of a Gaussian external force. We refer to [3, Section 4] for the modeling of (1.8) (see Eq. (4.2)).

1.3. Recent works

As mentioned at the end of Section 1.1 the first n-dimensional version of model (1.8) in the presence of a nonlinear sourcing term $f (u)$ was considered in [16] (Eq. (1.4)). Polynomial decay rates were obtained for regular initial data. Recently, Gomes Tavares et al. [11] consider model (1.4) in the context of frictional damping and without the presence of the force term $f (u)$ . The authors prove that the corresponding energy functional is squeezed by polynomial-like functions involving the power of the damping coefficient.

Sun and Yang in [17] considered the following model associated with (1.4)

\begin{aligned} u_{t t} - κ ϕ (‖ \nabla u ‖^{2}) Δ u + Δ^{2} u - M (‖ Δ u ‖^{2} + ‖ u_{t} ‖^{2}) Δ u_{t} + f (u) = h, \end{aligned}

(1.9)

with hinged boundary condition, where ϕ, M are functions of class

C^{1} (R^{+})

, with

M (s) > 0

for all

s \in R^{+}

, and h is an external force. The authors proved the existence of strong global attractors and their robustness on the perturbed parameter κ, where “strong” means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of a more regular space where the attractors belong. Very recently, Li et al. [14] complemented the results of [17] by proving the existence of a compact global attractor for problem (1.9) in the degenerate case

M (s) \equiv γ s^{q}

and with the dynamical system in the weak phase space topology. Later, Sun and Yang in [18] treated model (1.9) considering the presence of rotational forces

\begin{aligned} u_{t t} - Δ u_{t t} - κ ϕ (‖ \nabla u ‖^{2}) Δ u + Δ^{2} u - M (‖ Δ u ‖^{2} + ‖ u_{t} ‖^{2}) Δ u_{t} + f (u) = h . \end{aligned}

The existence of strong attractors has been established. The results obtained by Sun and Yang [17,18] are important works in the context of energy models, but they left open the degenerate case that corresponds to the natural and intrinsic form of the Balakrishnan-Taylor damping coefficient given by

M (s) \equiv γ s^{q}

Recently, Gomes Tavares et al. [10] treated model (1.4) considering frictional damping with a nonlocal fractional coefficient

\begin{aligned} γ {[‖ (- Δ)^{α} u ‖^{2} + ‖ u_{t} ‖^{2}]}^{q} u_{t}, α \in [0, 1] \end{aligned}

The authors prove the existence of a compact global attractor for the restriction of

α \in [0, 1)

. The case

α = 1

was considered with a Krasovskii’s dissipation given by

k (‖ Δ u ‖^{2} + ‖ u_{t} ‖^{2}) u_{t}

, where

k (\cdot)

is a bounded Lipschitz function on

R^{+}

such that

k = 0

[0, 1]

and

k (s)

is strictly increasing for

s > 1

. The existence of a non-compact attractor has been established. More recently, Gomes Tavares et al. [9], through the use of new nonlinear multipliers, proved the property of asymptotic compactness in the case

α = 1

, and consequently, the existence of a compact global attractor was established for

α = 1

. Also, recently, considering the damping energy in problem (1.4) of the form

\begin{aligned} [‖ Δ u ‖^{θ} + q ‖ u_{t} ‖^{ρ}] (- Δ)^{δ} u_{t}, θ ⩾ 1, q ⩾ 0, ρ > 0, 0 ⩽ δ ⩽ 1. \end{aligned}

Zhou and Sun [20] showed the well-posedness of weak solutions and the energy goes to zero as t goes to infinity when

q = 0

. When

q > 0

0 < δ ⩽ 1

, due to the degeneracy of the damping coefficient, the authors only proved the well-posedness of strong solutions to the problem. In the case

q = 0

, one can also refer to [6]. Also, more recently, Bezerra et al. [4,5] considered the context of energy damping for a class of plate models with non-constant material density. Results of stability and existence of global attractor were obtained.

Motivated by the aforementioned works, we propose to complement the work of Sun and Yang [18] in the open point left by the authors. We consider the degenerate case $M (s) \equiv γ s^{q}$ . To overcome the difficulty generated by the degenerate coefficient, a new class of nonlinear multipliers as set in [9] were used to establish the asymptotic compactness of the dynamic system associated with problem (1.1)-(1.3).

1.4. Organization of work

Our paper is organized as follows: in Section 2 we introduce the main assumptions and discuss the well-posedness to the problem (1.1)–(1.3). In Section 3 we establish our main result, Theorem 3.4, which guarantees the existence of a compact global attractor for (1.1)–(1.3).

2. Well-posedness

2.1. Functional spaces and assumptions

Let us begin by introducing some notation that will be used throughout this work. We denote $V_{0} = L^{2} (Ω)$ , $V_{1} = H_{0}^{1} (Ω)$ , and

\begin{aligned} V_{m} = {\begin{cases} H^{2} (Ω) \cap H_{0}^{1} (Ω) & if m = 2, \\ {u \in H^{m} (Ω) \cap H_{0}^{1} (Ω); Δ u \in H_{0}^{1} (Ω)} & if m = 3, 4. \end{cases} \end{aligned}

Here the notation

(\cdot, \cdot)

stands for

L^{2}

-inner product and

‖ \cdot ‖_{p}

denotes

L^{p}

-norm. For simplicity for

p = 2

‖ \cdot ‖_{2} = ‖ \cdot ‖

. Thus,

‖ \nabla \cdot ‖

and

‖ Δ \cdot ‖

represent the norms in

V_{1}

and

V_{2}

, respectively. Let’s denote

\begin{aligned} M_{γ} = I - γ Δ . \end{aligned}

The dynamics of the system generated by problem (1.1)–(1.3) will be studied on the following weak phase space

H_{γ} = V_{2} \times D (M_{γ}^{1 / 2})

with inner product and norm defined by

\begin{aligned} {⟨ (u^{1}, v^{1}), (u^{2}, v^{2}) ⟩}_{H_{γ}} = (Δ u^{1}, Δ u^{2}) + γ (\nabla v^{1}, \nabla v^{2}) + (v^{1}, v^{2}) \end{aligned}

and

\begin{aligned} {‖ (u, v) ‖}_{H_{γ}}^{2} = ‖ Δ u ‖^{2} + γ ‖ \nabla v ‖^{2} + ‖ v ‖^{2}, (u, v) \in H_{γ} . \end{aligned}

Note that

D (M_{γ}^{1 / 2}) \equiv V_{1} = H_{0}^{1} (Ω)

D (M_{γ}^{- 1 / 2}) \equiv V_{- 1} = H^{- 1} (Ω)

for

γ > 0

and

D (M_{γ}^{1 / 2}) \equiv V_{0} = L^{2} (Ω)

in the limiting case

γ \to 0

Denoting by $λ_{1} > 0$ the first eigenvalue of the bi-harmonic operator $Δ^{2}$ with hinged boundary condition, then

\begin{aligned} λ_{1} ‖ u ‖^{2} ⩽ ‖ Δ u ‖^{2}, λ_{1}^{1 / 2} ‖ \nabla u ‖^{2} ⩽ ‖ Δ u ‖^{2}, \forall u \in V_{2} . \end{aligned}

Let

U (t) = (u (t), u_{t} (t))

. The energy

E (t) = E (U (t))

related to problem (1.1)–(1.3) is given by

\begin{aligned} E (t) = \frac{1}{2} [{‖ U (t) ‖}_{H_{γ}}^{2} + κ {‖ \nabla u (t) ‖}^{2}] + \int_{Ω} [\hat{f} (u (t)) - h u (t)] d x, \end{aligned}

(2.1)

where

\hat{f} (u) = \int_{0}^{u} f (τ) d τ

. Assumption 2.1.

With respect to the forcing terms h and f we assume that $h \in L^{2} (Ω)$ , $f \in C^{1} (R)$ , and

\begin{aligned} | f^{'} (u) | ⩽ C_{f^{'}} (1 + | u |^{ρ}), u \in R, \end{aligned}

(2.2)

\begin{aligned} - C_{f} - \frac{c_{f}}{2} | u |^{2} ⩽ \hat{f} (u) ⩽ f (u) u + \frac{c_{f}}{2} | u |^{2}, u \in R, \end{aligned}

(2.3)

for some constants

C_{f}, C_{f^{'}} > 0

c_{f} \in [0, λ_{1})

and growth exponent

ρ ⩽ \frac{4}{n - 4}

for

n ⩾ 5

2.2. Generation of a semigroup

Setting $U = (u, v)$ with $v = u_{t}$ , we then rewrite the original problem (1.1)–(1.3) as the following equivalent first order problem

\begin{aligned} {\begin{cases} U_{t} = A U + B (U), t > 0, \\ U (0) = (u_{0}, u_{1}) := U_{0}, \end{cases} \end{aligned}

(2.4)

where

A : D (A) \subset H_{γ} \to H_{γ}

is a linear operator defined by

\begin{aligned} A U = {(\begin{matrix} v \\ - M_{γ}^{- 1} Δ^{2} u \end{matrix})}^{⊥}, U \in D (A) = {U \in H_{γ} | \begin{matrix} v \in V_{2}, \\ M_{γ}^{- 1} Δ^{2} u \in V_{1} \end{matrix}}, \end{aligned}

(2.5)

and

B : H_{γ} \to H_{γ}

is the nonlinear operator

\begin{aligned} B (U) = {(\begin{matrix} 0 \\ M_{γ}^{- 1} [κ Δ u + ‖ U (t) ‖_{H_{γ}}^{2 q} Δ v - f (u) + h] \end{matrix})}^{⊥}, U = (u, v) \in H_{γ} . \end{aligned}

(2.6)

Thus, the well-posedness result for (2.4), and consequently for the system (1.1)–(1.3), reads as follows: Theorem 2.1 Well-posedness

Let Assumption 2.1 be in force and $q ⩾ \frac{1}{2}$ . Then, we have:

I.
If $U_{0} \in H_{γ}$ , then there exists $T_{max} > 0$ such that problem ( 2.4 ) has a unique mild solution $U \in C ([0, T_{max}), H_{γ})$ , which is given by
$\begin{aligned} U (t) = e^{A t} U_{0} + \int_{0}^{t} e^{A (t - s)} B (U (s)) d s, t \in [0, T_{max}) . \end{aligned}$
(2.7)

II.
If $U_{0} \in D (A)$ , then the above mild solution U is regular one.
III.
In both cases, we have that $T_{max} = + \infty$ .

Proof.
The proof is based on four steps stated bellow. In Step 1 we will show that the operator $A : D (A) \subset H_{γ} \to H_{γ}$ given in (2.5) is an infinitesimal generator of a $C_{0}$ – semigroup of contractions on $H_{γ}$ . This is proved by showing that A is dissipative and maximal, and application of the Lumer-Phillips Theorem ([15], Theorem 1.4.3). In Step 2 we show that the operator $B : H_{γ} \to H_{γ}$ is locally lipschtz. Step 1 and Step 2 guarantee the existence of local solution ([15], Theorem 6.1.4). The existence of global solution is guaranteed in Step 3.

Step 1. The operator A defined in (2.5) is the infinitesimal generator of a $C_{0}$ -semigroup in $H_{γ}$ . Indeed, we take arbitrary element $U \in D (A)$ . Then
$\begin{aligned} {⟨ A U (t), U (t) ⟩}_{H_{γ}} = (Δ v, Δ u) + (M_{γ} [M_{γ}^{- 1} (- Δ^{2} u)], v) = 0. \end{aligned}$
Which shows that A is dissipative. To show that A is maximal we need to prove that $R (I - A) = H_{γ}$ , where $R (I - A)$ is the range of $I - A$ . Indeed, let $U^{} = (u^{}, v^{}) \in H_{γ}$ , and consider the equation $(I - A) U = U^{}$ which, written in components, reads
$\begin{aligned} u - v = u^{}, \\ M_{γ} v + Δ^{2} u = M_{γ} v^{} . \end{aligned}$
(2.8)
Substituting $u = v + u^{}$ in the second equations of (2.8), we obtain
$\begin{aligned} M_{γ} v + Δ^{2} v = M_{γ} v^{} - Δ^{2} u^{} =: w^{} \in V_{2}^{^{'}} . \end{aligned}$
(2.9)
Since the corresponding weak formulation is
$\begin{aligned} a (v, w) = \int_{Ω} w^{} w d x, \forall w \in V_{2}, \end{aligned}$
where
$\begin{aligned} a (v, w) := \int_{Ω} [v w + γ \nabla v \nabla w + Δ v Δ w] d x, \end{aligned}$
by the Lax–Milgram Theorem we can conclude that problem (2.9) admits a unique solution $v \in V_{2}$ . Then we deduce from the second equation of (2.8) that $M_{γ}^{- 1} Δ^{2} u = v^{} - v \in V_{1}$ . This implies that $R (I - A) = H_{γ}$ . Therefore, A is maximal monotone and due to Lummer-Phillips Theorem A is a infinitesimal generator of a $C_{0}$ – semigroup of contractions on $H_{γ}$ .

Step 2. The operator $B : H_{γ} \to H_{γ}$ given in (2.6) is locally Lipschitz. Indeed, let us take $R > 0$ and $U^{1} = (u^{1}, u_{t}^{1})$ , $U^{2} = (u^{2}, u_{t}^{2})$ such that $‖ U^{1} ‖_{H_{γ}}, ‖ U^{2} ‖_{H_{γ}} ⩽ R$ . Denoting $w = u^{1} - u^{2}$ , we have
$\begin{aligned} {‖ B (U^{1}) - B (U^{2}) ‖}_{H_{γ}} & = ‖ M_{γ}^{- 1 / 2} (Δ w + [{‖ U^{1} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{1} - {‖ U^{2} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{2}] - [f (u^{1}) - f (u^{2})]) ‖ \\ = {‖ Δ w + [{‖ U^{1} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{1} - {‖ U^{2} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{2}] - [f (u^{1}) - f (u^{2})] ‖}_{V_{- 1}} . \end{aligned}$
(2.10)
Now, let’s estimate the terms on the right hand side of the above equality. First we have
$\begin{aligned} \int_{Ω} Δ w φ d x ⩽ ‖ Δ w ‖ ‖ φ ‖ ⩽ ‖ U^{1} - U^{2} ‖_{H_{γ}} ‖ φ ‖, \forall φ \in V_{1} . \end{aligned}$
Using that
$\begin{aligned} \int_{Ω} [{‖ U^{1} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{1} - {‖ U^{2} ‖}_{H_{γ}}^{2 q} Δ u_{t}^{2}] φ d x & = \int_{Ω} {‖ U^{1} ‖}_{H_{γ}}^{2 q} Δ w_{t} φ d x \\ + \int_{Ω} [{‖ U^{1} ‖}_{H_{γ}}^{2 q} - {‖ U^{2} ‖}_{H_{γ}}^{2 q}] Δ u_{t}^{2} φ d x, \end{aligned}$
we have
$\begin{aligned} \int_{Ω} {‖ U^{1} (t) ‖}_{H_{γ}}^{2 q} Δ w_{t} φ d x ⩽ R^{2 q} ‖ \nabla w_{t} ‖ ‖ \nabla φ ‖ ⩽ R^{2 q} ‖ U^{1} - U^{2} ‖_{H_{γ}} ‖ \nabla φ ‖, \forall φ \in V_{1} \end{aligned}$
and, from Mean Value Theorem (MVT), there exists $θ \in (0, 1)$ such that
$\begin{aligned} \int_{Ω} [{‖ U^{1} ‖}_{H_{γ}}^{2 q} - {‖ U^{2} ‖}_{H_{γ}}^{2 q}] Δ u_{t}^{2} φ d x \\ ⩽ 2 q {θ {‖ U^{1} (t) ‖}_{H_{γ}} + (1 - θ) {‖ U^{2} (t) ‖}_{H_{γ}}}^{2 q - 1} ‖ U^{1} - U^{2} ‖_{H_{γ}} ‖ \nabla u_{t}^{2} ‖ ‖ \nabla φ ‖ \\ ⩽ \frac{2 q R^{2 q}}{γ^{1 / 2}} ‖ U^{1} - U^{2} ‖_{H_{γ}} ‖ \nabla φ ‖, \forall φ \in V_{1} . \end{aligned}$
From MVT, (2.2), Hölder’s inequality with $\frac{ρ}{2 (ρ + 1)} + \frac{1}{2 (ρ + 1)} + \frac{1}{2} = 1$ , and embedding $V_{2} ↪ L^{2 (ρ + 1)} (Ω)$ with $‖ v ‖_{2 (ρ + 1)} ⩽ C_{| Ω |} ‖ Δ v ‖$ , we get
$\begin{aligned} \int_{Ω} (f (u^{1}) - f (u^{2})) φ d x & ⩽ C_{f^{'}} \int_{Ω} [1 + {(| u^{1} | + | u^{2} |)}^{ρ}] | w | | φ | d x \\ ⩽ 2 C_{f^{'}} {[| Ω | + {‖ u^{1} ‖}_{2 (ρ + 1)}^{2 (ρ + 1)} + {‖ u^{2} ‖}_{2 (ρ + 1)}^{2 (ρ + 1)}]}^{\frac{ρ}{2 (ρ + 1)}} ‖ w ‖_{2 (ρ + 1)} ‖ φ ‖ \\ ⩽ 2^{ρ + 1} C_{f^{'}} [| Ω |^{\frac{ρ}{2 (ρ + 1)}} + {‖ u^{1} ‖}_{2 (ρ + 1)}^{ρ} + {‖ u^{2} ‖}_{2 (ρ + 1)}^{ρ}] ‖ w ‖_{2 (ρ + 1)} ‖ φ ‖ \\ ⩽ 2^{ρ + 1} C_{f^{'}} C_{| Ω |} [| Ω |^{\frac{ρ}{2 (ρ + 1)}} + C_{| Ω |}^{ρ} ({‖ Δ u^{1} ‖}^{ρ} + {‖ Δ u^{2} ‖}^{ρ})] ‖ Δ w ‖ ‖ φ ‖ \\ ⩽ 2^{ρ + 1} C_{f^{'}} C_{| Ω |} [| Ω |^{\frac{ρ}{2 (ρ + 1)}} + 2 C_{| Ω |}^{ρ} R^{ρ}] ‖ U^{1} - U^{2} ‖_{H_{γ}} ‖ φ ‖ . \end{aligned}$
Returning to (2.10), there exists a constant $C_{R} > 0$ such that
$\begin{aligned} ‖ B (U^{1}) - B (U^{2}) ‖_{H_{γ}} ⩽ C_{R} ‖ U^{1} - U^{2} ‖_{H_{γ}} . \end{aligned}$
(2.11)
This proves items $I .$ and $I I .$ of the Theorem 2.1.

Step 3. Remains to check that both mild and regular solutions are globally defined, that is, $T_{max} = + \infty$ . Indeed, in order we define
$\begin{aligned} ω := 1 - \frac{c_{f}}{λ_{1}} > 0. \end{aligned}$
Next, multiplying the Eq. (1.1) by $u_{t}$ and integrating over Ω we get
$\begin{aligned} \frac{d}{d t} E (U (t)) = - {‖ U (t) ‖}_{H_{γ}}^{2 q} {‖ \nabla u_{t} (t) ‖}^{2} ⩽ 0. \end{aligned}$
(2.12)
From (2.12), we have
$\begin{aligned} E (U (t)) ⩽ E (U (0)), \forall t \in [0, T_{max}) . \end{aligned}$
(2.13)
Now, from (2.3) and using immersion $V_{2} ↪ V_{0}$ , we have
$\begin{aligned} \int_{Ω} \hat{f} (u) d x ⩾ - \frac{c_{f}}{2} {‖ u (t) ‖}^{2} - C_{f} | Ω | ⩾ - \frac{c_{f}}{2 λ_{1}} {‖ Δ u (t) ‖}^{2} - C_{f} | Ω | . \end{aligned}$
(2.14)
On the other hand, from Hölder and Young inequalities, and again immersion $V_{2} ↪ V_{0}$ , we have
$\begin{aligned} \int_{Ω} h u d x ⩽ ‖ h ‖ ‖ u (t) ‖ ⩽ ‖ h ‖ \frac{1}{λ_{1}^{1 / 2}} ‖ Δ u (t) ‖ ⩽ \frac{1}{ω λ_{1}} ‖ h ‖^{2} + \frac{ω}{4} {‖ Δ u (t) ‖}^{2} . \end{aligned}$
(2.15)
Using the definition of $E (U (t))$ and inequalities (2.14) and (2.15) we get
$\begin{aligned} E (U (t)) ⩾ \frac{1}{2} {‖ M_{γ}^{1 / 2} u_{t} (t) ‖}^{2} + \frac{ω}{4} {‖ Δ u (t) ‖}^{2} - C_{f} | Ω | - \frac{1}{ω λ_{1}} ‖ h ‖^{2} ⩾ \frac{ω}{4} {‖ U (t) ‖}_{H_{γ}}^{2} - ω_{0}, \end{aligned}$
(2.16)
where $ω_{0} = [C_{f} | Ω | + \frac{1}{ω λ_{1}} ‖ h ‖_{2}^{2}]$ . Hence, from (2.16) and (2.13), we obtain
$\begin{aligned} {‖ U (t) ‖}_{H_{γ}}^{2} ⩽ \frac{4}{ω} E (U_{0}) + \frac{4 ω_{0}}{ω}, \forall t \in [0, T_{max}) . \end{aligned}$
(2.17)
Estimate (2.17) implies that any (mild or strong) solution is globally bounded in time. Therefore, from Pazy [15, Theorem 1.4] we conclude that $T_{max} = + \infty$ . This completes the proof of the Theorem 2.1.

2.3. Continuous dependence

Theorem 2.2.
Let Assumption 2.1 be in force, $h \in L^{2} (Ω)$ and $q ⩾ \frac{1}{2}$ . If $U^{1} (t) = (u^{1} (t), u_{t}^{1} (t))$ and $U^{2} (t) = (u^{2} (t), u_{t}^{2} (t))$ are two mild or strong solutions of problem (1.1)–(1.3) corresponding to initial data $U^{1} (0) = U_{0}^{1} = (u_{0}^{1}, u_{1}^{1})$ , $U^{2} (0) = U_{0}^{2} = (u_{0}^{2}, u_{1}^{2})$ , respectively. Then, there exists a positive constant $C = C (‖ U_{0}^{1} ‖_{H_{γ}}, ‖ U_{0}^{2} ‖_{H_{γ}})$ such that
$\begin{aligned} {‖ U^{1} (t) - U^{2} (t) ‖}_{H_{γ}} ⩽ e^{C t} {‖ U_{0}^{1} - U_{0}^{2} ‖}_{H_{γ}}, t \in [0, T] . \end{aligned}$
(2.18)
Proof.
From (2.7) we have
$\begin{aligned} U^{1} (t) - U^{2} (t) = e^{A t} [U_{0}^{1} - U_{0}^{2}] + \int_{0}^{t} e^{A (t - s)} [B (U^{1} (s)) - B (U^{2} (s))] d s . \end{aligned}$
(2.19)
Then, from (2.19) and (2.11) there exists a constant $C = C (‖ U_{0}^{1} ‖_{H_{γ}}, ‖ U_{0}^{2} ‖_{H_{γ}}) > 0$ such that
$\begin{aligned} {‖ U^{1} (t) - U^{2} (t) ‖}_{H_{γ}} ⩽ ‖ U_{0}^{1} - U_{0}^{2} ‖_{H_{γ}} + C \int_{0}^{t} {‖ U^{1} (s) - U^{2} (s) ‖}_{H_{γ}} d s . \end{aligned}$
Therefore, applying Gronwall’s lemma we get (2.18).
3. Global attractor

The general theory of dissipative dynamic systems can be found in [1,7,8,12,19]. To study the long-time behavior of solutions of the system (1.1)–(1.3) we will use the theory presented by Chueshov and Lasiecka in [7,8]. More specifically, we will show the existence of a global attractor proving that the system is dissipative and asymptotically smooth.

The well-posedness of problem (1.1)–(1.3) given by Theorems 2.1 and 2.2 implies that the evolution operator $S_{t} : H_{γ} \to H_{γ}$ defined by

\begin{aligned} S_{t} (U_{0}) = (U (t)), t ⩾ 0, \end{aligned}

where

U = (u, u_{t})

is the unique weak solution of the system (1.1)–(1.3), defines a nonlinear

C_{0}

-semigroup which is locally Lipschitz continuous on the phase space

H_{γ}

. Therewith the dynamics of problem (1.1)–(1.3) can be studied through the continuous dynamical system

(H_{γ}, S_{t})

A bounded set $B$ is an absorbing set for $(H_{γ}, S_{t})$ if for any bounded set $B \subset H_{γ}$ , there exists $t_{B} = t_{B} (B) ⩾ 0$ such that

\begin{aligned} S_{t} B \subset B, \forall t ⩾ t_{B}, \end{aligned}

which characterizes

(H_{γ}, S_{t})

as a dissipative dynamical system.

The dynamical system $(H_{γ}, S_{t})$ is asymptotically smooth if for any bounded positive invariant set $B \subset H_{γ}$ , there exists a compact set $K \subset \bar{B}$ , such that

\begin{aligned} dist (S_{t} B, K) \to 0 as t \to \infty . \end{aligned}

A bounded closed set $A$ is a global attractor for $(H_{γ}, S_{t})$ if, it is positive invariant, that is, $S_{t} A = A$ , $\forall t ⩾ 0$ , and uniformly attracting, that is, for any bounded set $B \subset H_{γ}$ ,

\begin{aligned} dist (S_{t} B, A) = sup_{x \in S_{t} B} inf_{y \in A} d (x, y) \to 0 as t \to \infty . \end{aligned}

Let $N$ be the set of stationary points of the dynamical system $(H_{γ}, S_{t})$ :

\begin{aligned} N = {v \in X : S (t) v = v for all t ⩾ 0} . \end{aligned}

We define the unstable manifold

M^{u} (N)

emanating from set

N

as a set of all

y \in H_{γ}

such that there exists a full trajectory

Υ = {u (t) : t \in R}

with the properties

\begin{aligned} u (0) = y and lim_{t \to - \infty} {dist}_{H_{γ}} (u (t), N) = 0. \end{aligned}

The dynamical system $(H_{γ}, S_{t})$ is said to be gradient if there exists a strict Lyapunov function for $(H_{γ}, S_{t})$ on the whole phase space $H_{γ}$ .

It is well known that the properties of dissipativity and asymptotic smoothness are critical for proving existence of global attractors. In fact, the following result is well known [7,8].

Theorem 3.1 Theorem 2.3, [7]

Let $S_{t}$ be a dissipative semigroup defined on a metric space H. Then $S (t)$ has a compact global attractor in H if and only if it is asymptotically smooth in H.

Theorem 3.2 Theorem 7.5.6, [8]

Let a dynamical system $(X, S_{t})$ possess a compact global attractor $A$ . Assume that there exists a strict Lyapunov function on $A$ . Then $A = M^{u} (N)$ . Moreover, the global attractor $A$ consists of full trajectories $Υ = {u (t) : t \in R}$ with the properties

\begin{aligned} lim_{t \to - \infty} {dist}_{X} (u (t), N) = 0 a n d lim_{t \to + \infty} {dist}_{X} (u (t), N) = 0. \end{aligned}

Proposition 3.3.
Assume that the assumptions of Theorem 2.1 hold. Then, $Φ : H_{γ} \to R$ given by
$\begin{aligned} Φ (U) := \frac{1}{2} ‖ U ‖_{H_{γ}}^{2} + \frac{κ}{2} ‖ \nabla u ‖^{2} + \int_{Ω} \hat{f} (u (x)) d x - \int_{Ω} h (x) u (x) d x \end{aligned}$
is a strict Lyapunov functional for the dynamical system $(H_{γ}, S_{t})$ . Consequently, $(H_{γ}, S_{t})$ is a gradient dynamical system.
Proof.
Let us define $Φ := E$ . From (2.12) one sees that the mapping
$\begin{aligned} t \mapsto E (u (t), u_{t} (t)) = Φ (S_{t} U_{0}) \end{aligned}$
is non-increasing for every $U_{0} := (u_{0}, u_{1}) \in H_{γ}$ . We assume that u is sufficiently regular for multiplying the Eq. (1.1) by $u_{t}$ and integrating over $Ω \times [0, t]$ and obtain
$\begin{aligned} Φ (S_{t} U_{0}) + \int_{0}^{t} {‖ U (τ) ‖}_{H_{γ}}^{2 q} {‖ \nabla u_{t} (τ) ‖}^{2} d τ = Φ (U_{0}), t > 0, \end{aligned}$
(3.1)
for every $U_{0} \in H_{γ}$ . From (3.1), it easily concludes that
$\begin{aligned} Φ (S_{t} U_{0}) = Φ (U_{0}) \Rightarrow U_{0} \in N_{λ}, t > 0, \end{aligned}$
where $N_{λ}$ is the set of stationary points of the dynamical system $(H_{γ}, S_{t})$ . Since we know that
$\begin{aligned} U_{0} \in N_{λ} \Leftrightarrow S_{t} (U_{0}) = U_{0}, t > 0, \end{aligned}$
then Φ is a strict Lyapunov functional for the dynamical system $(H, S_{t})$ .

We are now in a position to state our main result of this work.
Theorem 3.4 Global attractor

Let Assumptions of Theorem 2.2 be valid with $q ⩾ 1$ and $2^{q} q < 4$ . Then, we have:

I.
(Global attractor). The associate semigroup $S_{t}$ of problem ( 1.1 )–(1.3) has a compact global attractor $A_{γ}$ in $H_{γ}$ .
II.
(Characterization). The global attractor $A_{γ}$ is precisely the unstable manifold $A_{γ} = M^{u} (N)$ emanating from the set of stationary solution $N$ . In addition, $A_{γ}$ consist of full trajectories $Υ = {U (t) = (u (t), u_{t} (t)) : t \in R}$ such that
$\begin{aligned} lim_{t \to - \infty} {dist}_{H_{γ}} (U (t), N) = 0 a n d lim_{t \to + \infty} {dist}_{H_{γ}} (U (t), N) = 0. \end{aligned}$

Proof.
The proof of item $I .$ is based on two propositions stated below. Proposition 3.5 guarantees that $(H_{γ}, S_{t})$ is dissipative and Proposition 3.8 shows that $(H_{γ}, S_{t})$ has the asymptotic smoothness property. Then the existence of a global attractor follows from Theorem 3.1. The proof of item $I I .$ follows from Proposition 3.3 as a direct application of Theorem 3.2.

These results are detailed in the following three sections.
3.1. Dissipativity

Proposition 3.5 Dissipativity

Under the hypotheses of Theorem 2.1, the dynamical system $(H_{γ}, S_{t})$ is dissipative. Moreover, there exists a positively bounded absorbing set $B \subset H_{γ}$ .

Proof.
Let us begin by fixing an arbitrary bounded set $B \subset H_{γ}$ and consider the solutions of problem (1.1)–(1.3) given by $S_{t} (U_{0}) = U (t)$ with $U_{0} = (u_{0}, u_{1}) \in B$ .

We define the perturbed energy
$\begin{aligned} \tilde{E} (U (t)) = E (U (t)) + ω_{0}, with~ ω_{0} = \frac{1}{ω λ_{1}} ‖ h ‖^{2} + C_{f} | Ω | . \end{aligned}$
From (2.16), we have
$\begin{aligned} \tilde{E} (U (t)) ⩾ \frac{ω}{4} {‖ U (t) ‖}_{H_{γ}}^{2} . \end{aligned}$
(3.2)
By multiplying the Eq. (1.1) by $u_{t}$ and integrating over $Ω \times [t, t + 1]$ , we have
$\begin{aligned} \int_{t}^{t + 1} {‖ U (s) ‖}_{H_{γ}}^{2 q} {‖ \nabla u_{t} (s) ‖}^{2} d s = E (U (t)) - E (U (t + 1)) = D {(t)}^{2} . \end{aligned}$
(3.3)
Using that $‖ U (t) ‖_{H_{γ}}^{2 q} ⩾ γ ‖ \nabla u_{t} ‖^{2 q}$ we get
$\begin{aligned} {‖ U (t) ‖}_{H_{γ}}^{2 q} {‖ \nabla u_{t} (t) ‖}^{2} ⩾ γ^{q} {‖ \nabla u_{t} (t) ‖}^{2 (q + 1)} . \end{aligned}$
Hence, returning to (3.3), we have
$\begin{aligned} \int_{t}^{t + 1} {‖ \nabla u_{t} (s) ‖}^{2 (q + 1)} d s ⩽ \frac{1}{γ^{q}} D {(t)}^{2} . \end{aligned}$
From Hölder inequality with $\frac{q}{q + 1} + \frac{1}{q + 1} = 1$ , we have
$\begin{aligned} \int_{t}^{t + 1} {‖ \nabla u_{t} (s) ‖}^{2} d s ⩽ {[\int_{t}^{t + 1} {‖ \nabla u_{t} (s) ‖}^{2 (q + 1)} d s]}^{\frac{1}{q + 1}} ⩽ \frac{1}{γ^{\frac{q}{q + 1}}} D {(t)}^{\frac{2}{q + 1}} . \end{aligned}$
(3.4)
From Mean Value Theorem (MVT) there exist $t_{1} \in [t, t + \frac{1}{4}]$ , $t_{2} \in [t + \frac{3}{4}, t + 1]$ such that
$\begin{aligned} {‖ \nabla u_{t} (t_{i}) ‖}^{2} ⩽ \frac{4}{γ^{\frac{q}{q + 1}}} D {(t)}^{\frac{2}{q + 1}}, for~ i = 1, 2. \end{aligned}$
(3.5)
Next, multiplying the Eq. (1.1) by u and integrating over $Ω \times [t_{1}, t_{2}]$ we get
$\begin{aligned} \int_{t_{1}}^{t_{2}} \int_{Ω} [| Δ u |^{2} + κ | \nabla u |^{2} + f (u) u - h u] d x d s = \int_{t_{1}}^{t_{2}} {‖ M_{γ}^{1 / 2} u_{t} (s) ‖}^{2} d s + \overset{2}{\sum_{i = 1}} I_{i}, \end{aligned}$
(3.6)
where
$\begin{aligned} I_{1} & = \int_{t_{1}}^{t_{2}} \int_{Ω} {‖ U (s) ‖}_{H_{γ}}^{2 q} Δ u_{t} u d x d s, \\ I_{2} & = - {[\int_{Ω} u_{t} (s) u (s) d x]}_{t_{1}}^{t_{2}} - γ {[\int_{Ω} \nabla u_{t} (s) \nabla u (s) d x]}_{t_{1}}^{t_{2}} . \end{aligned}$
From (2.3), definition of $\tilde{E}$ , and (3.4) we get
$\begin{aligned} \int_{t_{1}}^{t_{2}} \tilde{E} (U (s)) d s ⩽ ω_{0} + \frac{1 + λ_{1}^{\frac{1}{2}}}{λ_{1}^{\frac{1}{2}} γ^{\frac{q}{q + 1}}} D {(t)}^{\frac{2}{q + 1}} + \overset{2}{\sum_{i = 1}} I_{i} . \end{aligned}$
(3.7)
The terms $I_{1}$ and $I_{2}$ can be estimated as follows. First, from Hölder inequality, (2.17), (3.2), and (3.3), we have
$\begin{aligned} I_{1} & ⩽ \int_{t_{1}}^{t_{2}} \int_{Ω} {‖ U (s) ‖}_{H_{γ}}^{q} ‖ \nabla u_{t} ‖ {‖ U (s) ‖}_{H_{γ}}^{q} | \nabla u | d x d s \\ ⩽ {\int_{t}^{t + 1} {‖ U (s) ‖}_{H_{γ}}^{2 q} {‖ \nabla u_{t} (s) ‖}^{2} d s}^{1 / 2} {\int_{t}^{t + 1} {‖ U (s) ‖}_{H_{γ}}^{2 q} {‖ \nabla u (s) ‖}^{2} d s}^{1 / 2} \\ ⩽ C_{B} D (t) sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s))^{1 / 2} \\ ⩽ C_{ϵ, B} D {(t)}^{2} + ε sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) . \end{aligned}$
Now, using immersion $V_{2} ↪ V_{1} ↪ V_{0}$ , (3.2), and (3.5) we have
$\begin{aligned} I_{2} & ⩽ C_{λ_{1}} \overset{2}{\sum_{i = 1}} ‖ \nabla u_{t} (i) ‖ sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s))^{1 / 2} \\ ⩽ \frac{2 C_{λ_{1}}}{γ^{\frac{q}{2 (q + 1)}}} D {(t)}^{\frac{1}{q + 1}} sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s))^{1 / 2} \\ ⩽ \frac{C_{λ_{1}}^{2}}{γ^{\frac{q}{q + 1}} ε} D {(t)}^{\frac{2}{q + 1}} + ε sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) . \end{aligned}$
Substituting $I_{1}$ and $I_{2}$ into (3.7) we get
$\begin{aligned} \int_{t_{1}}^{t_{2}} \tilde{E} (U (s)) d s ⩽ ω_{0} + C_{ϵ} D {(t)}^{\frac{2}{q + 1}} + C_{ϵ, B} D {(t)}^{2} + 2 ε sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) . \end{aligned}$
(3.8)
Again, using the MVT, there exists $τ \in [t_{1}, t_{2}]$ such that
$\begin{aligned} \int_{t_{1}}^{t_{2}} \tilde{E} (U (s)) d s = \tilde{E} (τ) (t_{2} - t_{1}) ⩾ \frac{1}{2} \tilde{E} (U (t + 1)) . \end{aligned}$
(3.9)
On the other hand, from (3.3), we have
$\begin{aligned} \tilde{E} (U (t)) = \tilde{E} (U (t + 1)) + D {(t)}^{2} . \end{aligned}$
(3.10)
From (3.8), (3.9), (3.10), and using that $\tilde{E} (U (t)) = sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s))$ , we obtain
$\begin{aligned} sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) ⩽ 2 ω_{0} + 2 C_{ϵ} D {(t)}^{\frac{2}{q + 1}} + (1 + 2 C_{ϵ, B}) D {(t)}^{2} + 4 ε sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) . \end{aligned}$
Taking $ε = \frac{1}{8}$ we get
$\begin{aligned} sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) ⩽ 4 ω_{0} + D {(t)}^{\frac{2}{q + 1}} [4 C_{ϵ} + 2 (1 + 2 C_{ϵ, B}) D {(t)}^{\frac{2 q}{q + 1}}] . \end{aligned}$
(3.11)
From the definition of $D (t)$ and using that $E (U (t)) ⩽ E (U_{0})$ for all $t ⩾ 0$ , we infer that
$\begin{aligned} 4 C_{ε} + 2 (1 + 2 C_{ε, B}) D {(t)}^{\frac{2 q}{q + 1}} ⩽ 4 C_{ε} + 2^{\frac{2 q + 1}{q + 1}} (1 + 2 C_{ε, B}) {| E (U_{0}) |}^{\frac{q}{q + 1}} ⩽ C_{ε, B} . \end{aligned}$
Thus, (3.11) can be rewritten as
$\begin{aligned} sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s)) ⩽ 4 ω_{0} + C_{ε, B} D {(t)}^{\frac{2}{q + 1}} . \end{aligned}$
Using that $(a + b)^{q + 1} ⩽ 2^{q} (a^{q + 1} + b^{q + 1})$ , we have
$\begin{aligned} sup_{t ⩽ s ⩽ t + 1} \tilde{E} (U (s))^{q + 1} ⩽ 2^{3 q + 2} ω_{0}^{q + 1} + {\tilde{C}}_{ε, B} [\tilde{E} (U (t)) - \tilde{E} (U (t + 1))] . \end{aligned}$
Thus, applying Nakao’s lemma, we obtain
$\begin{aligned} \tilde{E} (U (t)) ⩽ {[{\tilde{C}}_{ε, B}^{- 1} q (t - 1)^{+} + \tilde{E} (U_{0})^{- q}]}^{- \frac{1}{q}} + 2^{\frac{3 q + 2}{q + 1}} ω_{0}, 0 ⩽ t < T, \end{aligned}$
where $(t - 1)^{+} = max {t - 1, 0}$ . From (3.2) we get
$\begin{aligned} {‖ U (t) ‖}_{H_{γ}}^{2} ⩽ \frac{4}{ω} {[{\tilde{C}}_{ε, B}^{- 1} q (t - 1)^{+} + \tilde{E} (U_{0})^{- q}]}^{- \frac{1}{q}} + \frac{2^{\frac{5 q + 4}{q + 1}} ω_{0}}{ω}, 0 ⩽ t < T . \end{aligned}$
(3.12)
This estimate show that the set
$\begin{aligned} B = {\bar{B (0, R)}}^{H_{γ}}, R = \frac{2^{\frac{5 q + 4}{q + 1}} ω_{0}}{ω} \end{aligned}$
is a absorbing set for $S_{t}$ in $H_{γ}$ . Therefore the dynamical system $(H_{γ}, S_{t})$ is dissipative.
Corollary 3.6 Polynomial decay rate

We assume the hypotheses of Theorem 2.1 with $h \equiv 0$ and $C_{f} = 0$ in Assumption (2.3). Then there exists a positive constants $C = C_{‖ U_{0} ‖_{H_{γ}}}$ such that the energy $E (t)$ definid in (2.1) has the following decay rates

\begin{aligned} E (U (t)) ⩽ C (1 + t)^{- \frac{1}{q}} . \end{aligned}

(3.13)

Proof.

Let $U_{0} \in B \subset H$ . Using that $h = 0$ and $C_{f} = 0$ , we have $ω_{0} = 0$ and $\tilde{E} = E$ . Thus, from the inequality (3.12), we obtain that

\begin{aligned} E (U (t)) ⩽ \frac{4}{ω} {[{\tilde{C}}_{ε, B}^{- 1} q (t - 1)^{+} + E (U_{0})^{- q}]}^{- \frac{1}{q}} \end{aligned}

Therefore, (3.13) follows directly from the inequality above.

3.2. Asymptotic smoothness

The proof that the system $(H_{γ}, S_{t})$ is asymptotically smooth is set out in Proposition 3.8 below. The result is obtained by a direct application of the compactness criterion presented in [8, Theorem 7.1.11].

Theorem 3.7 Theorem 7.1.11, [8]

Let $(X, S_{t})$ be a dynamical system on a complete metric space X endowed with a metric d. Assume that for any bounded positively invariant set B in X and for any $ε > 0$ there exists $T = T_{B, ε}$ such that

\begin{aligned} d (S_{T} (y_{1}), S_{T} (y_{2})) = ε + Ψ_{ε, B, T} (y_{1}, y_{2}), y_{i} \in B, \end{aligned}

where

Ψ_{ε, B, T} (y_{1}, y_{2})

is a functional defined on

B \times B

such that

\begin{aligned} lim inf_{m \to \infty} lim inf_{n \to \infty} Ψ_{ε, B, T} (y_{n}, y_{m}) = 0 \end{aligned}

for every sequence

y_{n}

from B. Then

(X, S_{t})

is an asymptotically smooth dynamical system.

Proposition 3.8 Asymptotic smoothness

Under assumptions of Theorem 2.2 with $2^{q} q < 4$ , then the dynamical system $(H_{γ}, S_{t})$ generated by mild solutions to (1.1)–(1.3) is asymptotically smooth.

Proof.

The proof is a direct application of the compactness criterion presented in [8, Theorem 7.1.11]. Let B a bounded set in $H_{γ}$ and $S_{t} (U_{0}^{i}) = (u^{i}, u_{t}^{i})$ be two mild solutions of problem (1.1)-(1.3) with initial data $U_{0}^{i} \equiv (u_{0}^{i}, u_{1}^{i}) \in B$ , $i = 1, 2$ . Let $T > 0$ and $w (t) = u^{1} (t) - u^{2} (t)$ . The variable $w = u^{1} - u^{2}$ satisfies $(w (0), w_{t} (0)) = U_{0}^{1} - U_{0}^{2}$ and

\begin{aligned} (I - γ Δ) w_{t t} - κ Δ w + Δ^{2} w - \frac{1}{2} Π_{1} Δ w_{t} - \frac{1}{2} Π_{2} [Δ u_{t}^{1} + Δ u_{t}^{2}] + F (w) = 0, \end{aligned}

(3.14)

where

\begin{aligned} F (w) = f (u^{1}) - f (u^{2}) and Π_{j} (t) := {[{‖ U^{1} ‖}_{H_{γ}}^{2}]}^{q} + (- 1)^{j - 1} {[{‖ U^{2} ‖}_{H_{γ}}^{2}]}^{q}, j = 1, 2. \end{aligned}

We set

\begin{aligned} E_{w} (t) := \frac{1}{2} [{‖ (w (t), w_{t} (t)) ‖}_{H_{γ}}^{2} + κ ‖ \nabla w (t) ‖^{2}] . \end{aligned}

Multiplying the equation (3.14) by

w_{t}

and integrating over

Ω \times [τ, T]

, we get

\begin{aligned} E_{w} (T) - E_{w} (τ) + \frac{1}{2} \int_{τ}^{T} Π_{1} (s) ‖ \nabla w_{t} ‖^{2} d s + \frac{1}{2} \int_{τ}^{T} Π_{2} (s) \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w_{t} d x d s \\ = \int_{τ}^{T} \int_{Ω} F (w) w_{t} d x . d s . \end{aligned}

(3.15)

Let

ξ_{θ} = θ ‖ U^{1} ‖_{H_{γ}}^{2} + (1 - θ) ‖ U^{2} ‖_{H_{γ}}^{2}

, from MVT there exists

θ \in (0, 1)

such that

\begin{aligned} \frac{1}{2} Π_{2} (s) \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w_{t} d x & = \frac{q γ}{2} [ξ_{θ}]^{q - 1} {[‖ \nabla u_{t}^{1} ‖^{2} - ‖ \nabla u_{t}^{2} ‖^{2}]}^{2} \\ + \underset{J_{1} (t)}{\underset{⏟}{\frac{q}{2} [ξ_{θ}]^{q - 1} [‖ u_{t}^{1} ‖^{2} - ‖ u_{t}^{2} ‖^{2}] \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w_{t} d x}} \\ + \underset{J_{2} (t)}{\underset{⏟}{\frac{q}{2} [ξ_{θ}]^{q - 1} [‖ Δ u^{1} ‖^{2} - ‖ Δ u^{2} ‖^{2}] \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w_{t} d x}} . \end{aligned}

Returning to (3.15) we get

\begin{aligned} E_{w} (T) - E_{w} (τ) + \frac{1}{2} \int_{τ}^{T} Π_{1} (s) {‖ \nabla w_{t} (s) ‖}^{2} d s + \frac{q γ}{2} \int_{τ}^{T} [ξ_{θ}]^{q - 1} {[{‖ \nabla u_{t}^{1} ‖}^{2} - {‖ \nabla u_{t}^{2} ‖}^{2}]}^{2} d s \\ = \int_{τ}^{T} [- J_{1} (s) - J_{2} (s) + \int_{Ω} F (w) w_{t} d x] d s . \end{aligned}

(3.16)

For simplicity, in the following we will use

C_{B}

to denote several constants that depend on B. Using that

a^{2} - b^{2} = (a - b) (a + b)

\begin{aligned} ‖ u_{t}^{j} ‖ ⩽ {‖ U^{j} ‖}_{H_{γ}} and ‖ \nabla u_{t}^{j} ‖ ⩽ \frac{1}{γ^{1 / 2}} {‖ U^{j} ‖}_{H_{γ}}, j = 1, 2, \end{aligned}

and Young’s inequality, we can estimate the integrals of the terms

J_{1}

and

J_{2}

as follows

\begin{aligned} - \int_{τ}^{T} J_{1} (s) d s \\ ⩽ \frac{q}{2} \int_{0}^{T} [‖ U^{1} ‖_{H_{γ}}^{2} + ‖ U^{2} ‖_{H_{γ}}^{2}] [‖ u_{t}^{1} ‖ + ‖ u_{t}^{2} ‖] | ‖ u_{t}^{1} ‖ - ‖ u_{t}^{2} ‖ | [‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖] ‖ \nabla w_{t} ‖ d s \\ ⩽ \frac{q}{2 γ^{1 / 2}} \int_{τ}^{T} {[‖ U^{1} ‖_{H_{γ}}^{2} + ‖ U^{2} ‖_{H_{γ}}^{2}]}^{q - 1} {[‖ U^{1} ‖_{H_{γ}} + ‖ U^{2} ‖_{H_{γ}}]}^{2} ‖ w_{t} ‖ ‖ \nabla w_{t} ‖ d s \\ ⩽ \frac{q}{γ^{1 / 2}} \int_{0}^{T} {[‖ U^{1} ‖_{H_{γ}}^{2} + ‖ U^{2} ‖_{H_{γ}}^{2}]}^{q} ‖ w_{t} ‖ ‖ \nabla w_{t} ‖ d s \\ ⩽ C_{B} T sup_{0 ⩽ t ⩽ T} ‖ w_{t} (t) ‖ \end{aligned}

and

\begin{aligned} \int_{τ}^{T} J_{2} (s) d s \\ = \frac{q}{2} \int_{τ}^{T} [ξ_{θ}]^{q - 1} [‖ Δ u^{1} ‖ + ‖ Δ u^{2} ‖] [‖ Δ u^{1} ‖ - ‖ Δ u^{2} ‖] [{‖ \nabla u_{t}^{1} ‖}^{2} - {‖ \nabla u_{t}^{2} ‖}^{2}] d s \\ ⩽ \frac{q γ}{2} \int_{τ}^{T} [ξ_{θ}]^{q - 1} {[{‖ \nabla u_{t}^{1} ‖}^{2} - {‖ \nabla u_{t}^{2} ‖}^{2}]}^{2} d s \\ + \frac{q}{8 γ} \int_{τ}^{T} [ξ_{θ}]^{q - 1} {[‖ Δ u^{1} ‖ + ‖ Δ u^{2} ‖]}^{2} {[‖ Δ u^{1} ‖ - ‖ Δ u^{2} ‖]}^{2} d s \\ ⩽ \frac{q γ}{2} \int_{τ}^{T} [ξ_{θ}]^{q - 1} {[{‖ \nabla u_{t}^{1} ‖}^{2} - {‖ \nabla u_{t}^{2} ‖}^{2}]}^{2} d s \\ + \frac{q}{4 γ} \int_{τ}^{T} {[{‖ U^{1} ‖}_{H_{γ}}^{2} + ‖ U^{2} ‖_{H_{γ}}^{2}]}^{q - 1} [{‖ U^{1} ‖}_{H_{γ}}^{2} + {‖ U^{2} ‖}_{H_{γ}}^{2}] ‖ Δ w ‖^{2} d s \\ ⩽ \frac{q γ}{2} \int_{τ}^{T} [ξ_{θ}]^{q - 1} {[{‖ \nabla u_{t}^{1} ‖}^{2} - {‖ \nabla u_{t}^{2} ‖}^{2}]}^{2} d s + \frac{2^{q} q}{8 γ} \int_{τ}^{T} Π_{1} (s) ‖ Δ w ‖^{2} d s . \end{aligned}

Finally, from MVT, Hölder inequality with

\frac{ρ}{2 (ρ + 1)} + \frac{ρ + 2}{4 (ρ + 1)} + \frac{ρ + 2}{4 (ρ + 1)} = 1

, immersions

V_{1} ↪ L^{\frac{4 (ρ + 1)}{ρ + 2}} (Ω)

with

‖ v ‖_{\frac{4 (ρ + 1)}{ρ + 2}} ⩽ {\tilde{C}}_{| Ω |} ‖ \nabla v ‖

V_{2} ↪ L^{2 (ρ + 1)} (Ω)

with

‖ v ‖_{2 (ρ + 1)} ⩽ C_{| Ω |} ‖ Δ v ‖

, and Young inequality with

\frac{1}{2 (q + 1)} + \frac{2 q + 1}{2 (q + 1)} = 1

, we obtain

\begin{aligned} \int_{τ}^{T} \int_{Ω} F (w) w_{t} d x d s \\ = - \int_{τ}^{T} \int_{Ω} \int_{0}^{1} f^{'} (θ u^{1} + (1 - θ) u^{2}) d θ w w_{t} d x d s \\ ⩽ C_{f^{'}} \int_{τ}^{T} \int_{Ω} [1 + {(| u^{1} | + | u^{2} |)}^{ρ}] | w | | w_{t} | d x d s \\ ⩽ 2^{ρ + 1} C_{f^{'}} \int_{τ}^{T} {[| Ω | + {‖ u^{1} ‖}_{2 (ρ + 1)}^{2 (ρ + 1)} + {‖ u^{2} ‖}_{2 (ρ + 1)}^{2 (ρ + 1)}]}^{\frac{ρ}{2 (ρ + 1)}} ‖ w ‖_{\frac{4 (ρ + 1)}{ρ + 2}} ‖ w_{t} ‖_{\frac{4 (ρ + 1)}{ρ + 2}} d s \\ ⩽ 2^{ρ + 1} C_{f^{'}} {\tilde{C}}_{| Ω |}^{2} \int_{0}^{T} {[| Ω | + C_{| Ω |}^{2 (ρ + 1)} (‖ Δ u^{1} ‖^{2 (ρ + 1)} + ‖ Δ u^{2} ‖^{2 (ρ + 1)})]}^{\frac{ρ}{2 (ρ + 1)}} ‖ \nabla w ‖ ‖ \nabla w_{t} ‖ d s \\ ⩽ C_{B} T sup_{0 ⩽ t ⩽ T} ‖ \nabla w (t) ‖ . \end{aligned}

Substituting the last three inequalities in (3.16), we get

\begin{aligned} E_{w} (T) + \frac{1}{2} \int_{τ}^{T} Π_{1} (s) ‖ \nabla w_{t} ‖^{2} d s & ⩽ E_{w} (τ) + \frac{2^{q} q}{8 γ} \int_{τ}^{T} Π_{1} (s) ‖ Δ w ‖^{2} d s \\ + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} . \end{aligned}

(3.17)

On the other hand, multiplying the equation (3.14) by

\frac{2^{q} q}{8 γ} Π_{1} (t) w

and integrating over

Ω \times [τ, T]

, we have

\begin{aligned} \frac{2^{q} q}{8 γ} \int_{τ}^{T} Π_{1} (s) [‖ Δ w ‖^{2} + κ ‖ \nabla w ‖^{2}] d s = \frac{2^{q} q}{8} \int_{τ}^{T} Π_{1} (s) ‖ \nabla w_{t} ‖^{2} d s + \overset{5}{\sum_{i = 1}} \int_{τ}^{T} L_{i} (s) d s, \end{aligned}

(3.18)

where

\begin{aligned} L_{1} (t) & = - \frac{2^{q} q}{8 γ} Π_{1} (t) \int_{Ω} F (w) w d x, \\ L_{2} (t) & = \frac{2^{q} q}{8 γ} Π_{1} (t) ‖ w_{t} ‖^{2}, \\ L_{3} (t) & = - \frac{2^{q} q}{16 γ} Π_{1} {(t)}^{2} \int_{Ω} \nabla w_{t} \nabla w d x, \\ L_{4} (t) & = - \frac{2^{q} q}{16 γ} Π_{2} (t) Π_{1} (t) \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w (t) d x, \\ L_{5} (t) & = \frac{2^{q} q}{8 γ} Π_{1} (t) \frac{d}{d t} [\int_{Ω} (w_{t} w + γ \nabla w_{t} \nabla w) d x] . \end{aligned}

Hence, combining (3.17) and (3.18), we obtain that

\begin{aligned} E_{w} (T) + (\frac{1}{2} - \frac{2^{q} q}{8}) \int_{τ}^{T} Π_{1} (s) ‖ \nabla w_{t} ‖^{2} d s & ⩽ E_{w} (τ) + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} \\ + \overset{5}{\sum_{i = 1}} \int_{τ}^{T} L_{i} (s) d s . \end{aligned}

(3.19)

Still, using that

\begin{aligned} Π_{1} (t) ⩾ γ {‖ \nabla u_{t}^{1} ‖}^{2 q} + γ {‖ \nabla u_{t}^{2} ‖}^{2 q} ⩾ \frac{γ}{2^{2 q - 1}} {[‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖]}^{2 q} ⩾ \frac{γ}{2^{2 q - 1}} ‖ \nabla w_{t} ‖^{2 q}, \end{aligned}

substituting in (3.19), we have

\begin{aligned} E_{w} (T) + γ C_{q} \int_{τ}^{T} ‖ \nabla w_{t} ‖^{2 (q + 1)} d s & ⩽ E_{w} (τ) + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} \\ + \overset{5}{\sum_{i = 1}} \int_{τ}^{T} L_{i} (s) d s . \end{aligned}

(3.20)

where

C_{q} = \frac{1}{2^{q}} (\frac{1}{2^{q}} - \frac{q}{4})

. The integrals of the terms

L_{1} (t), \dots, L_{5} (t)

are estimated as follows. First, from MVT, Hölder inequality with

\frac{ρ}{2 (ρ + 1)} + \frac{ρ + 2}{4 (ρ + 1)} + \frac{ρ + 2}{4 (ρ + 1)} = 1

, immersions

V_{1} ↪ L^{\frac{4 (ρ + 1)}{ρ + 2}} (Ω)

and

V_{2} ↪ L^{2 (ρ + 1)} (Ω)

, we obtain

\begin{aligned} \int_{τ}^{T} L_{1} (s) d s ⩽ C_{B} \int_{τ}^{T} ‖ w ‖_{\frac{4 (ρ + 1)}{ρ + 2}}^{2} d s ⩽ C_{B} \int_{τ}^{T} ‖ \nabla w ‖^{2} d s ⩽ C_{B, T} sup_{0 ⩽ t ⩽ T} ‖ \nabla w (t) ‖ . \end{aligned}

By direct calculations it is easy to see that

\begin{aligned} \int_{τ}^{T} L_{2} (s) d s ⩽ C_{B, T} sup_{0 ⩽ t ⩽ T} ‖ w_{t} (t) ‖^{2}, \\ \int_{τ}^{T} L_{3} (s) d s ⩽ C_{B, T} sup_{0 ⩽ t ⩽ T} ‖ \nabla w (t) ‖, \\ \int_{τ}^{T} L_{4} (s) d s ⩽ C_{B, T} sup_{0 ⩽ t ⩽ T} ‖ \nabla w (t) ‖ . \end{aligned}

Now let’s estimate the last term

L_{5}

. Integrating by parts we get

\begin{aligned} \int_{τ}^{T} L_{5} (s) d s = \underset{L_{5, 1}}{\underset{⏟}{\frac{2^{q} q}{8 γ} Π_{1} (T) \int_{Ω} (w (T) w_{t} (T) + γ \nabla w (T) \nabla w_{t} (T)) d x}} \\ \underset{L_{5, 2}}{- \underset{⏟}{\frac{2^{q} q}{8 γ} Π_{1} (τ) \int_{Ω} (w (τ) w_{t} (τ) + γ \nabla w (τ) \nabla w_{t} (τ)) d x}} \\ \underset{L_{5, 3}}{+ \underset{⏟}{\frac{2^{q} q}{8 γ} \int_{τ}^{T} Π_{1}^{^{'}} (s) \int_{Ω} (w w_{t} + γ \nabla w \nabla w_{t}) d x d s}} \end{aligned}

The terms

L_{5, 1}

and

L_{5, 2}

can be easily estimated as follows

\begin{aligned} L_{5, 1} & ⩽ C_{B} [‖ Δ w (T) ‖ + ‖ w (T) ‖] ‖ w_{t} (T) ‖ ⩽ C_{B} sup_{0 ⩽ t ⩽ T} ‖ w_{t} (t) ‖, \\ L_{5, 2} & ⩽ C_{B} [‖ Δ w (τ) ‖ + ‖ w (τ) ‖] ‖ w_{t} (τ) ‖ ⩽ C_{B} sup_{0 ⩽ t ⩽ T} ‖ w_{t} (t) ‖ . \end{aligned}

To estimate the integral of the term

L_{5, 3}

first note that

\begin{aligned} Π_{1}^{^{'}} (t) = q {[{‖ U^{1} ‖}_{H_{γ}}^{2} + ‖ U^{2} ‖_{H_{γ}}^{2}]}^{q - 1} [\frac{d}{d t} {‖ U^{1} ‖}_{H_{γ}}^{2} + \frac{d}{d t} {‖ U^{2} ‖}_{H_{γ}}^{2}] . \end{aligned}

On the other hand, with each solution

U^{i}

satisfies

\begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ U^{i} ‖}_{H_{γ}}^{2} + {‖ U^{i} ‖}_{H_{γ}}^{2 q} ‖ \nabla u_{t}^{i} ‖^{2} = κ (Δ u^{i}, u_{t}^{i}) - (f (u^{i}), u_{t}^{i}) + (h, u_{t}^{i}) . \end{aligned}

Then, from Assumption (2.2) and Hölder inequality, we have

\begin{aligned} \frac{d}{d t} {‖ U^{i} ‖}_{H_{γ}}^{2} ⩽ C . \end{aligned}

Thus, we have the following estimate for the integral of

L_{5, 3} (t)

\begin{aligned} \int_{τ}^{T} L_{5, 3} (s) d s ⩽ C_{B} sup_{0 ⩽ t ⩽ T} ‖ w_{t} (t) ‖ . \end{aligned}

Substituting the integral of the terms

L_{1} (t), \dots, L_{5} (t)

in (3.20), we obtain

\begin{aligned} E_{w} (T) + γ C_{q} \int_{τ}^{T} ‖ \nabla w_{t} ‖^{2 (q + 1)} d s ⩽ E_{w} (τ) + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} . \end{aligned}

(3.21)

Integrating (3.21) from 0 to T, we get

\begin{aligned} T E_{w} (T) + γ C_{q} \int_{0}^{T} \int_{τ}^{T} ‖ \nabla w_{t} ‖^{2 (q + 1)} d s d τ & ⩽ \int_{0}^{T} E_{w} (τ) d τ \\ + T C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} . \end{aligned}

(3.22)

Next, multiplying the equation (3.14) by w and integrating over

Ω \times [0, T]

, we obtain that

\begin{aligned} \int_{0}^{T} E_{w} (τ) d τ = 2 γ \int_{0}^{T} ‖ \nabla w_{t} ‖^{2} d τ + \overset{5}{\sum_{i = 1}} \int_{τ}^{T} {\tilde{L}}_{i} (s) d s, \end{aligned}

(3.23)

where

\begin{aligned} {\tilde{L}}_{1} (t) & := 2 ‖ w_{t} ‖^{2}, \\ {\tilde{L}}_{2} (t) & := - \int_{Ω} F (w) w d x, \\ {\tilde{L}}_{3} (t) & := - \frac{1}{2} Π_{1} (t) \int_{Ω} \nabla w_{t} \nabla w d x, \\ {\tilde{L}}_{4} (t) & := - \frac{1}{2} Π_{2} (t) \int_{Ω} [\nabla u_{t}^{1} + \nabla u_{t}^{2}] \nabla w (t) d x, \\ {\tilde{L}}_{5} (t) & := \frac{d}{d t} [\int_{Ω} (w_{t} w + γ \nabla w_{t} \nabla w) d x] . \end{aligned}

Similarly, as the integrals of terms

L_{i}^{^{'}} s

were estimated, from (3.23), we obtain that

\begin{aligned} \int_{0}^{T} E_{w} (τ) d τ ⩽ 2 γ \int_{0}^{T} ‖ \nabla w_{t} ‖^{2} d τ + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} . \end{aligned}

(3.24)

Combining (3.22) and (3.24), we have

\begin{aligned} T E_{w} (T) + γ C_{q} \int_{τ}^{T} ‖ \nabla w_{t} ‖^{2 (q + 1)} d s \\ ⩽ 2 γ \int_{0}^{T} ‖ \nabla w_{t} ‖^{2} d τ + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}} . \end{aligned}

(3.25)

From Hölder inequality with

\frac{q}{q + 1} + \frac{1}{q + 1} = 1

and using (3.21) with

τ = 0

, we obtain

\begin{aligned} 2 γ \int_{0}^{T} ‖ \nabla w_{t} ‖^{2} d τ & ⩽ 2 (γ T)^{\frac{q}{q + 1}} {[γ \int_{0}^{T} ‖ \nabla w_{t} ‖^{2 (q + 1)} d τ]}^{\frac{1}{q + 1}} \\ ⩽ 2 (γ T)^{\frac{q}{q + 1}} {[\frac{1}{C_{q}} E_{w} (0) + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}}]}^{\frac{1}{q + 1}} \end{aligned}

Returning to (3.25), we obtain that

\begin{aligned} E_{w} (T) ⩽ \frac{2 γ^{\frac{q}{q + 1}}}{C_{q} T^{\frac{1}{q + 1}}} E_{w} (0)^{\frac{1}{q + 1}} + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}}^{\frac{1}{q + 1}} . \end{aligned}

Using that

\frac{1}{2} ‖ S_{T} U_{0}^{1} - S_{T} U_{0}^{2} ‖_{H_{γ}}^{2} ⩽ E_{w} (T)

and

E_{w} (0) ⩽ C_{B}

, we get

\begin{aligned} {‖ S_{T} U_{0}^{1} - S_{T} U_{0}^{2} ‖}_{H_{γ}}^{2} ⩽ \frac{C_{B}}{T^{\frac{1}{q + 1}}} + C_{B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t)) ‖}_{V_{1} \times V_{0}}^{\frac{1}{q + 1}} . \end{aligned}

Therefore, for any

ϵ > 0

there exists

T \equiv T_{ϵ, B} > 0

such that

\begin{aligned} {‖ S_{T} (U_{0}^{1}) - S_{T} (U_{0}^{2}) ‖}_{H_{γ}}^{2} ⩽ ϵ + C_{ϵ, B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t) ‖}_{V_{1} \times V_{0}}^{\frac{1}{q + 1}} . \end{aligned}

(3.26)

We define

Ψ_{B, ϵ, T} : B \times B \to R

\begin{aligned} Ψ_{B, ϵ, T} (U_{0}^{1}, U_{0}^{2}) := C_{ϵ, B, T} sup_{0 ⩽ t ⩽ T} {‖ (w (t), w_{t} (t) ‖}_{V_{1} \times V_{0}}^{\frac{1}{q + 1}} . \end{aligned}

Then, from (3.26), we have

\begin{aligned} {‖ S_{T} (U_{0}^{1}) - S_{T} (U_{0}^{2}) ‖}_{H_{γ}}^{2} ⩽ ϵ + Ψ_{B, ϵ, T} (U_{0}^{1}, U_{0}^{2}) . \end{aligned}

(3.27)

Now, given a sequence

{U_{0}^{n}}_{n \in N} \subset B

, since B is bounded and positive invariant, the corresponding solutions

S_{t} (U_{0}^{n}) = U^{n} (t)

of problem (1.1)–(1.3) are uniformly bounded in

H_{γ}

. Hence

\begin{aligned} {U^{n}} is bounded in C ([0, \infty); V_{2} \times V_{1}) . \end{aligned}

Thus, since

H_{γ} \equiv_{γ > 0} V_{2} \times V_{1} ↪ V_{1} \times V_{0}

compactly, there exists a subsequence

{U^{n_{k}}} = {(u^{n_{k}}, u^{n_{k}})}

which converge strongly in

C ([0, \infty), V_{1} \times V_{0})

. Therefore

\begin{aligned} lim_{k \to \infty} lim_{l \to \infty} Ψ_{ϵ, B, T} (U_{0}^{n_{k}}, U_{0}^{n_{l}}) \\ = lim_{k \to \infty} lim_{l \to \infty} C_{B, ϵ, T} sup_{0 ⩽ t ⩽ T} {‖ (u^{n_{k}} (t) - u^{n_{l}} (t), u_{t}^{n_{k}} (t) - u_{t}^{n_{l}} (t)) ‖}_{V_{1} \times V_{0}}^{\frac{1}{q + 1}} = 0. \end{aligned}

(3.28)

Therefore, from (3.27) and (3.28), applying [8, Theorem 7.1.11] this proves that

(H_{γ}, S_{t})

is an asymptotically smooth dynamical system.

Footnotes

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggestions which helped us to improve the original manuscript considerably.

Yanan Li is supported by the National Natural Science Foundation of China (No.12101155) and Heilongjiang Province Natural Science Foundation of China (No. LH2021A001). Vando Narciso is supported by the Fundect/CNPq Grant 15/2024.

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