Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term

Abstract

This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.

Keywords

Fractal dimension subcritical case Kirchhoff wave equation global attractor stabilizability inequality

1. Introduction

This article is motivated by recent studies on the so-called Kirchhoff wave equation with strong damping and memory in [17] $\begin{aligned} (1.1) & \{\begin{array}{ll} ε (t) u_{t t} - M (‖ \nabla u ‖^{2}) Δ u - Δ u_{t} - k (0) Δ u - \int_{0}^{\infty} k^{'} (s) Δ u (t - s) d s \\ = g (u) + h (x) & in Ω \times R^{+}, \\ u (x, t) = 0 & on \partial Ω \times R^{+}, \\ u (x, t) = u_{0} (x, t), u_{t} (x, t) = \partial_{t} u_{0} (x, t) & x \in Ω, t ⩽ 0, \end{array} \end{aligned}$ where Ω is a bounded domain in $R^{n}$ $(n \in N^{+})$ with smooth boundary $\partial Ω$ , $ε (t)$ is a time-dependent function, $M (s) = s^{\frac{m}{2}}$ for any $m ⩾ 1$ and $s \in R^{+}$ , $- Δ u_{t}$ is a strong damping, $k (\cdot)$ admits $k (\infty) > 0$ and $k^{'} (s) ⩽ 0$ , $g (u)$ is a nonlinear function, $h (x) \in L^{2} (Ω)$ is independent of time.

We make some assumptions as follows,

(H1) The time-dependent function $ε (t) \in C^{1} (R)$ is decreasing and satisfying $\begin{aligned} (1.2) & lim_{t \to + \infty} ε (t) = a \end{aligned}$ for some $0 < a < 1$ and there exists a constant $L > a > 0$ such that $\begin{aligned} (1.3) & sup_{t \in R} (| ε (t) | + | ε^{'} (t) |) ⩽ L . \end{aligned}$

(H2) $g (u) \in C^{1} (R)$ and there exists a constant $p_{2}$ that fulfills $p_{2} = 1$ when $n = 1$ , 2 and $p_{2} \in (1, \frac{n + 2}{n - 2})$ when $n ⩾ 3$ such that $\begin{aligned} (1.4) & | g^{'} (s) | ⩽ C_{2} (1 + | s |^{p_{2} - 1}), \\ (1.5) & | g (u) - g (v) | ⩽ (1 + | u |^{p_{2} - 1} + | v |^{p_{2} - 1}) | v - u | \end{aligned}$ and $\begin{aligned} (1.6) & \underset{| s | \to \infty}{lim inf} \frac{g (s)}{s} < λ_{1}, \end{aligned}$ where $λ_{1} > 0$ is the constant in the Poincaré inequality $λ_{1} ‖ u ‖^{2} ⩽ ‖ \nabla u ‖^{2}$ , $p_{*} = \frac{n + 2}{n - 2}$ is called to be critical exponent.

(H3) The memory kernel $μ (\cdot)$ in (1.11) satisfies $μ (s) \in C^{1} (R^{+}) \cap L^{1} (R^{+})$ $\begin{aligned} (1.7) & μ (s) ⩾ 0, μ^{'} (s) ⩽ 0, \\ (1.8) & \int_{0}^{\infty} μ (s) d s = δ_{1}, \\ (1.9) & μ^{'} (s) + δ_{2} μ (s) ⩽ 0 \end{aligned}$ for any $s \in R^{+}$ , where $δ_{1}$ and $δ_{2}$ satisfy $δ_{2} ⩾ δ_{1} > 0$ .

As in [6], we define $\begin{aligned} (1.10) & η^{t} (x, s) = u (x, t) - u (x, t - s) . \end{aligned}$

Taking the derivatives of (1.10) with respect to t and using assumption $μ (s) = - k^{'} (s)$ , $k (\infty) = 1$ , then we can rewrite (1.1) as $\begin{aligned} (1.11) & ε (t) u_{t t} - (1 + ‖ \nabla u ‖^{m}) Δ u - Δ u_{t} - \int_{0}^{\infty} μ (s) Δ η^{t} (x, s) d s = g (u) + h (x) \end{aligned}$ and $\begin{aligned} (1.12) & η_{t} = - η_{s} + u_{t} \end{aligned}$ with initial boundary value conditions $\begin{aligned} (1.13) & \{\begin{array}{ll} u (x, t) = 0, & x \in \partial Ω, t ⩾ 0, \\ η^{t} (x, s) = 0, & (x, s) \in \partial Ω \times R^{+}, t ⩾ 0, \\ u (x, 0) = u_{0} (x), & x \in Ω, \\ u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ η^{t} (x, 0) = 0, & x \in Ω, \\ η^{0} (x, s) = η_{0} (x, s), & (x, s) \in \partial Ω \times R^{+}, \end{array} \end{aligned}$ where $\begin{aligned} (1.14) & \{\begin{array}{ll} u_{0} (x) = u_{0} (x, 0), & x \in \partial Ω, \\ u_{1} (x) = \partial_{t} u_{0} (x, t) |_{t = 0}, & x \in \partial Ω, \\ η_{0} (x, s) = u_{0} (x, 0) - u_{0} (x, - s) & (x, s) \in Ω \times R^{+} . \end{array} \end{aligned}$

As an important tool to characterize the geometric properties and complexity of attractors, fractal dimenison has been widely investigated in various equations. Among them, the wave equations of Kirchhoff type are the important models which have been studied recently. In 1883, Kirchhoff [7] firstly introduced the equation $u_{t t} - M (‖ \nabla u ‖^{2}) Δ u = h$ to describe small vibrations of an elastic stretch string. Yang [18] got the existence of global attractor of the wave equations of Kirchhoff type with strong damping $u_{t t} - M (‖ \nabla u ‖^{2}) Δ u - Δ u_{t} + u + u_{t} + g (x, u) = f (x)$ in $H^{2} (Ω) \times H^{1} (Ω)$ and proved that they have finite fractal and Hausdorff dimensions. Besides, Chueshov [1] studied the well-posedness and long-time dynamics for Kirchhoff wave models with strong nonlinear damping $\partial_{t t} u - σ (‖ \nabla u ‖^{2}) Δ \partial_{t} u - ϕ (‖ \nabla u ‖^{2}) Δ u + f (u) = h (x)$ . Subsequently, in [8,9,11,21], the long-time dynamics of wave equations of Kirchhoff type with constant coefficients were studied separately. On the basis of Chueshov’s research [1], Yang and Ding [19] extended the results with strictly positive stiffness factors and supercritical nonlinearity, then further got the finite-dimensional global attractor in the natural energy space endowed with strong topology. Later on, Ding and Yang [4] obtained the existence of global and exponential attractors for the strongly damped wave equation of Kirchhoff type $u_{t t} - ϕ (x) (‖ \nabla u ‖^{2}) Δ u_{t} - ϕ (x) M (‖ \nabla u ‖^{2}) Δ u + f (u) = h (x)$ with supercritical nonlinearity in a weighted energy space. Furthermore, Yang and Li [20] found the pullback attractor is upper semicontinuous with respect to α for non-autonomous wave equation of Kirchhoff type $u_{t t} - M (‖ \nabla u ‖^{2}) Δ u + {(- Δ)}^{α} u_{t} + f (u) = g (x, t)$ .

In addition, Ma and Zhang [12] researched the same equation as in Chueshov [1] under suitable assumptions, the fractal dimension of the global attractor is infinite by using the method of $Z_{2}$ index. The wave equations of Kirchhoff type with a weak interior damping, under strict hyperbolism in [3,15,16] and mild degeneracies in [5,13] is analyzed. Recently, Lv and Lin [10] got the existence of the global attractor for a class of wave equations of Kirchhoff type with thermal effect and memory term $u_{t t} - Δ u_{t} - a (x) M (‖ \nabla u ‖^{2}) Δ u - \int_{0}^{\infty} k (s) u (t - s) d s + f (u) + μ a (x) Δ θ = h (x)$ , where $θ \in L^{\infty} (0, \infty; H) \cap L^{2} (0, T; H_{0}^{1})$ . Yang and Qin [17] proved the existence and regularity of the global attractor of problem (1.1). To the best of our knowledge, for the wave equations of Kirchhoff type with a strong damping and a memory term, there is no any result on the dimension of attractors in a weighted time-dependent space until now.

Our aim is to estimate the fractal dimension of the global attractor of problem (1.1). There are some known methods to estimate the fractal dimension of attractors: (1) Volume contraction method by calculating the Lyapunov exponents in the case that the semigroup is differentiable. (2) The method which is based on the squeezing property or the smoothing property of the difference of two solutions in the non-differentiable case. (3) The ℓ-trajectory method in the case of minimal regularity on the solutions. (4) The attractor is quasi-stable when it’s a compact attractor. In this paper, we will consider the fractal dimension of global attractor for problem (1.1) by verifing that the semigroup on the global attractor is quasi-stable.

The paper is organized as follows. In Section 2, we shall introduce some notations and basic lemmas and preliminaries. In Section 3, we shall prove our system $(H_{t}, S (t))$ is quasi-stable for the equation (1.1). In Section 4, we shall show the fractal dimension of the global attractor is finite, see Theorem 4.3.

2. Preliminaries

In this section, we will give some notations and results. Subsequently, we denote $V_{1} = L^{2} (Ω)$ , $V_{2} = H_{0}^{1} (Ω)$ , and $V_{3} = H_{0}^{1} (Ω) \cap H^{2} (Ω)$ . We also denote $‖ \cdot ‖ = ‖ \cdot ‖_{L^{2} (Ω)}$ then their norms as $‖ \cdot ‖_{V_{1}} = ‖ \cdot ‖$ , $‖ \cdot ‖_{V_{2}} = ‖ \nabla \cdot ‖$ and $‖ \cdot ‖_{V_{3}} = ‖ Δ \cdot ‖$ , respectively.

We suppose $M_{i} (i = 1, 2, 3)$ are $L_{μ}^{2}$ -weighted Hilbert spaces, which satisfy $\begin{aligned} (2.1) & M_{i} = L_{μ}^{2} (R^{+}; V_{i}) = {ξ (s) : R^{+} \to V_{i} ∣ {‖ ξ (s) ‖}_{M_{i}}^{2} < + \infty}, \end{aligned}$ and their inner products and norms are defined as $\begin{aligned} (2.2) & {(ξ (s), η (s))}_{M_{i}} = \int_{0}^{+ \infty} μ (s) {(ξ (s), η (s))}_{V_{i}} d s \end{aligned}$ and $\begin{aligned} (2.3) & {‖ ξ (s) ‖}_{M_{i}}^{2} = \int_{0}^{+ \infty} μ (s) {‖ ξ (s) ‖}_{V_{i}}^{2} d s, \end{aligned}$ respectively.

Moreover, our phase space is the weighted time-dependent space $H_{t} = V_{2} \times V_{1} \times M_{2}$ , which is equipped with the norm $\begin{aligned} (2.4) & {‖ (u, u_{t}, ξ) ‖}_{H_{t}}^{2} = ‖ \nabla u ‖^{2} + ε (t) ‖ u_{t} ‖^{2} + ‖ ξ ‖_{M_{2}}^{2} . \end{aligned}$

Throughout the paper, $C_{i}$ ( $i \in N$ ) stands for a universal positive constant.

Now, we recall some definitions and results related to the global attractor, which will be used in the present paper.

Definition 2.1.
Let ${S (t)}_{t ⩾ 0}$ be a semigroup on a metric space $(X, d)$ . A subset $A$ of X is called a global attractor for the semigroup, if $A$ is compact and enjoys the following properties:

(1) $A$ is invariant, that is, $S (t) A = A$ for all $t ⩾ 0$ ;

(2) $A$ attracts all bounded sets of X, that is, for any bounded subset B of X, $\begin{array}{r} d (S (t) B, A) \to 0, as t \to + \infty \end{array}$ where $d (B, A)$ is the Hausdorff semi-distance.
Definition 2.2.
Let M be a compact set in a metric space X. The fractal (box-counting) dimension ${dim}_{f} M$ of M is defined by $\begin{aligned} (2.5) & {dim}_{f} (M) = \underset{ε \to 0}{lim sup} \frac{log n (M, ε)}{log (1 / ε)}, \end{aligned}$ where $n (M, ε)$ is the minimal number of closed balls of the radius ε which cover the set M.

In what follows, we give a criterion of estimate the fractal dimension of global attractor.
Lemma 2.3 ([14]).

Let X, Y and Z be three reflexive Banach spaces with X compactly embedded in Y and $H = X \times Y \times Z$ . Consider the dynamical system $(H, S (t))$ given by an evolution operator $\begin{aligned} (2.6) & S (t) w = (u (t), u_{t} (t), η^{t}), w = (u_{0}, u_{1}, η_{0}) \in H, \end{aligned}$ where the functions u and η have regularity $\begin{aligned} (2.7) & u \in C (R^{+}; X) \cap C^{1} (R^{+}; Y), η \in C (R^{+}; Z) . \end{aligned}$ Then $(H, S (t))$ is quasi-stable on a set $B \subset H$ if there exists a compact seminorm $n_{X}$ on X and nonnegative scalar functions $a (t)$ and $c (t)$ , locally bounded in $[0, \infty)$ , and $b (t) \in L^{1} (R^{+})$ with ${lim}_{t \to \infty} b (t) = 0$ such that, $\begin{aligned} (2.8) & {‖ S (t) w^{1} - S (t) w^{2} ‖}_{H}^{2} ⩽ a (t) {‖ w^{1} - w^{2} ‖}_{H}^{2}, \end{aligned}$ and $\begin{aligned} (2.9) & {‖ S (t) w^{1} - S (t) w^{2} ‖}_{H}^{2} ⩽ b (t) {‖ w^{1} - w^{2} ‖}_{H}^{2} + c (t) sup_{0 < s < t} {[n_{X} (u^{1} (s) - u^{2} (s))]}^{2} \end{aligned}$ for any $w^{1}, w^{2} \in B$ . The inequality ( 2.9 ) is often called stabilizability inequality.

Let Assumptions (H1)–(H3) be in force, we directly state the following results in [17], which will be used in the present paper.

Lemma 2.4 ([17]).

Under the assumptions of $M (\cdot)$ , $ε (\cdot)$ , $μ (\cdot)$ , g and h in (H1)–(H3), the semigroup ${S (t)}_{t ⩾ 0}$ for problem ( 1.11 )–( 1.14 ) has a bounded absorbing set in $H$ . Moreover, $\begin{aligned} (2.10) & ε (t) ‖ u_{t} ‖^{2} + ‖ \nabla u ‖^{2} + ‖ η ‖_{M_{2}}^{2} ⩽ R_{1} . \end{aligned}$

Theorem 2.5 ([17]).

Under the assumptions of $M (\cdot)$ , $ε (\cdot)$ , $μ (\cdot)$ , g and h in (H1)–(H3), then the semigroup ${S (t)}_{t ⩾ 0}$ for problem ( 1.11 )–( 1.14 ) has a unique global attractor $A$ in the weighted time-dependent space $H$ .

3. A stabilizability like inequality

Lemma 3.1.
Suppose the assumptions (H1)–(H3) hold. Then $\begin{aligned} (3.1) & \int_{0}^{t} {‖ \nabla u_{t} (τ) ‖}^{2} d τ ⩽ C, \end{aligned}$ where $C > 0$ depends on $R_{1}$ and $‖ h ‖_{L^{2} (Ω)}$ . Proof.
Multiplying (1.11) by $u_{t}$ and using $G (u) = \int_{0}^{u} g (s) d s$ and (1.12), we arrive at $\begin{aligned} \frac{d}{d t} (\frac{1}{2} ε (t) ‖ u_{t} ‖^{2} + \frac{1}{2} ‖ \nabla u ‖^{2} + \frac{1}{m + 2} ‖ \nabla u ‖^{m + 2} + \frac{1}{2} ‖ η ‖_{M_{2}}^{2} - \int_{Ω} (G (u) + h u) d x) \\ (3.2) & - \frac{1}{2} ε^{'} (t) ‖ u_{t} ‖^{2} + ‖ \nabla u_{t} ‖^{2} + {(η, η_{s})}_{M_{2}} = 0 . \end{aligned}$

Define the energy functional corresponding to the system (1.11)–(1.14) is given by $\begin{aligned} (3.3) & E (t) = \frac{1}{2} ε (t) ‖ u_{t} ‖^{2} + \frac{1}{2} ‖ \nabla u ‖^{2} + \frac{1}{m + 2} ‖ \nabla u ‖^{m + 2} + \frac{1}{2} ‖ η ‖_{M_{2}}^{2} - \int_{Ω} (G (u) + h u) d x . \end{aligned}$

Noting that $ε (t)$ is decreasing and use assumption (H1), we derive $\begin{aligned} (3.4) & - \frac{1}{2} ε^{'} (t) ‖ u_{t} ‖^{2} > 0 . \end{aligned}$

From assumption (H3) and the definition of $M_{2}$ , we obtain $\begin{aligned} {(η, η_{s})}_{M_{2}} & = \int_{0}^{\infty} μ (s) \int_{Ω} \nabla η \nabla η_{s} d x d s = - \frac{1}{2} \int_{0}^{\infty} μ^{'} (s) ‖ \nabla η ‖^{2} d s \\ (3.5) & ⩾ \frac{δ_{2}}{2} \int_{0}^{\infty} μ (s) ‖ \nabla η ‖^{2} d s = \frac{δ_{2}}{2} ‖ η ‖_{M_{2}}^{2} . \end{aligned}$

Then inserting (3.3), (3.4) and (3.5) into (3.2), we conclude $\begin{aligned} (3.6) & \frac{d}{d t} E (t) + ‖ \nabla u_{t} ‖^{2} ⩽ 0 . \end{aligned}$

Integrating (3.6) from 0 to t, we derive $\begin{aligned} (3.7) & E (t) + \int_{0}^{t} {‖ \nabla u_{t} (τ) ‖}^{2} d τ ⩽ E (0) . \end{aligned}$

Consequently, from Lemma 2.4, we obtain (3.1) directly. □

Now we show an essential lemma to our proof of Theorem 4.3. Lemma 3.2.
Suppose the assumptions (H1)–(H3) hold. Given a bounded set $B \subset H$ , let $z_{i} = (u^{i}, {(u^{i})}_{t}, η_{i})$ be two weak solutions of problem ( 1.11 )–( 1.14 ) such that $z_{i} (0) = (u_{0}^{i}, u_{1}^{i}, η_{0 i}) \in B$ , $i = 1$ , 2. Then $\begin{aligned} {‖ z_{1} (t) - z_{2} (t) ‖}_{H_{t}}^{2} ⩽ & C e^{- κ t} {‖ z_{1} (0) - z_{2} (0) ‖}_{H_{t}}^{2} \\ (3.8) & + C \int_{0}^{t} e^{- κ (t - τ)} ({‖ u^{1} (τ) - u^{2} (τ) ‖}^{2} d τ \end{aligned}$ for any $t ⩾ 0$ , where $v > 0$ and $K_{B} > 0$ are constants.
Proof.
Let $z_{1} = (u^{1}, {(u^{1})}_{t}, η_{1})$ and $z_{2} = (u^{2}, {(u^{2})}_{t}, η_{2})$ be two weak solutions to problem (1.11)–(1.14), let $\bar{u} (t) = u^{1} (t) - u^{2} (t)$ and $\bar{η} = {\bar{η}}^{t} = η_{1}^{t} - η_{2}^{t}$ . Then we obtain $\begin{aligned} (3.9) & \{\begin{array}{l} ε (t) {\bar{u}}_{t t} - M_{1} (t) Δ \bar{u} - {\tilde{M}}_{1} (t) (\nabla (u^{1} + u^{2}), \nabla \bar{u}) Δ (u^{1} + u^{2}) - Δ \bar{u} - Δ {\bar{u}}_{t} \\ - \int_{0}^{\infty} μ (s) Δ \bar{η} (x, s) d s = g (u^{1}) - g (u^{2}), \\ {\bar{η}}_{t} = - {\bar{η}}_{s} + {\bar{u}}_{t}, \end{array} \end{aligned}$ where $\begin{aligned} M_{1} (t) = \frac{1}{2} (M ({‖ \nabla u^{1} ‖}_{L^{2} (Ω)}^{2}) + M ({‖ \nabla u^{2} ‖}^{2})) > 0, \\ {\tilde{M}}_{1} (t) = \frac{1}{2} \int_{0}^{1} M^{'} (λ {‖ \nabla u^{1} ‖}^{2} + (1 - λ) {‖ \nabla u^{2} ‖}^{2}) d λ > 0 \end{aligned}$ with initial condition $\begin{aligned} (3.10) & (\bar{u} (0), {\bar{u}}_{t} (0), {\bar{η}}^{0}) = z_{1} (0) - z_{2} (0) . \end{aligned}$

Mltiplying ${(3.9)}_{1}$ by ${\bar{u}}_{t}$ and ${(3.9)}_{2}$ by $\bar{η}$ , respectively, integrating the resulting equalities over Ω, we deduce $\begin{aligned} \frac{1}{2} \frac{d}{d t} (ε (t) ‖ {\bar{u}}_{t} ‖^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + ‖ \bar{η} ‖_{M_{2}}^{2} + M_{1} (t) ‖ \nabla \bar{u} ‖^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2}) \\ + ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}}^{2} + \frac{δ_{2}}{2} ‖ η ‖^{2} \\ = \frac{1}{2} (M^{'} (t) (\nabla u^{1}, \nabla u_{t}^{1}) + M^{'} (t) (\nabla u^{2}, \nabla u_{t}^{2})) ‖ \nabla \bar{u} ‖^{2} \\ + {\tilde{M}}_{1} (t) (\nabla (u_{t}^{1} + u_{t}^{2}), \nabla z) (\nabla (u^{1} + u^{2}), \nabla z) \\ + \frac{1}{2} \int_{0}^{1} M^{″} (λ {‖ \nabla u^{1} ‖}^{2} + (1 - λ) {‖ \nabla u^{2} ‖}^{2}) (λ (\nabla u^{1}, \nabla u_{t}^{1}) \\ + (1 - λ) (\nabla u^{1}, \nabla u_{t}^{2})) d λ {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2} \\ + (g (u^{1}) - g (u^{2}), {\bar{u}}_{t}) + \frac{1}{2} ε^{'} (t) ‖ {\bar{u}}_{t} ‖^{2} \\ (3.11) & ⩽ C (‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖) ‖ \nabla \bar{u} ‖^{2} + (g (u^{1}) - g (u^{2}), {\bar{u}}_{t}) . \end{aligned}$

When $1 ⩽ p_{2} < p^{*}$ , there exist a constant $ϵ > 0$ and a constant $δ : 0 < δ ≪ 1$ such that $H_{0}^{1 - δ} ↪ L^{p_{2} + 1}$ . By the interpolation theorem and the Young inequality $\begin{aligned} | (g (u^{1}) - g (u^{2}), {\bar{u}}_{t}) | \\ ⩽ C \int_{Ω} ({| u^{1} |}^{p_{2} - 1} + {| u^{2} |}^{p_{2} - 1} + 1) | \bar{u} ‖ {\bar{u}}_{t} | d x \\ ⩽ C ‖ \bar{u} ‖ ‖ {\bar{u}}_{t} ‖ + C ({‖ u^{1} ‖}_{p_{2} + 1}^{p_{2} - 1} + {‖ u^{2} ‖}_{p_{2} + 1}^{p_{2} - 1}) ‖ \bar{u} ‖_{p_{2} + 1} ‖ {\bar{u}}_{t} ‖_{p_{2} + 1} \\ ⩽ C ‖ \bar{u} ‖ ‖ {\bar{u}}_{t} ‖ + C ‖ \bar{u} ‖_{H_{0}^{1 - δ}} ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}} \\ ⩽ C ‖ \bar{u} ‖ ‖ {\bar{u}}_{t} ‖ + C ‖ \bar{u} ‖^{δ} ‖ \bar{u} ‖_{H_{0}^{1}}^{1 - δ} ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}} \\ (3.12) & ⩽ ϵ ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}}^{2} + ϵ^{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + C ‖ \bar{u} ‖^{2} . \end{aligned}$

Inserting (3.12) into (3.11), we derive $\begin{aligned} \frac{1}{2} \frac{d}{d t} (ε (t) ‖ {\bar{u}}_{t} ‖^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + ‖ \bar{η} ‖_{M_{2}}^{2} + M_{1} (t) ‖ \nabla \bar{u} ‖^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2}) \\ + (1 - ϵ) ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}}^{2} + \frac{δ_{2}}{2} ‖ η ‖^{2} \\ (3.13) & ⩽ ϵ^{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + C (‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖) ‖ \nabla \bar{u} ‖^{2} + C ‖ \bar{u} ‖^{2} . \end{aligned}$

Similarly, multiplying ${(3.9)}_{1}$ by $\bar{u}$ and ${(3.9)}_{2}$ by $\bar{η}$ , respectively, integrating the resulting equalities over Ω, we obtain $\begin{aligned} \frac{d}{d t} (ε (t) ({\bar{u}}_{t}, \bar{u}) + \frac{1}{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2}) - (ε^{'} (t) \bar{u_{t}}, \bar{u}) - ε (t) ‖ {\bar{u}}_{t} ‖^{2} + M_{1} (t) ‖ \nabla \bar{u} ‖^{2} \\ + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + {(\bar{η}, \bar{u})}_{M_{2}} \\ (3.14) & = (g (u^{1}) - g (u^{2}), \bar{u}) . \end{aligned}$

By using the Young inequality and assumption (H3), we arrive at there exists a constant $α_{0} > 0$ such that $\begin{aligned} | (ε^{'} (t) ({\bar{u}}_{t}, \bar{u}) | ⩽ - ε^{'} (t) ‖ \bar{u_{t}} ‖^{2} - \frac{ε^{'} (t)}{4} ‖ \bar{u} ‖^{2} \\ | {(\bar{η}, \bar{u})}_{M_{2}} | ⩽ α_{0} ‖ \bar{η} ‖_{M_{2}}^{2} + \frac{1}{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2} \end{aligned}$ and $\begin{array}{r} (g (u^{1}) - g (u^{2}), \bar{u}) ⩽ ϵ ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + C ‖ \bar{u} ‖^{2} . \end{array}$ Then we deduce $\begin{aligned} \frac{d}{d t} (ε (t) ({\bar{u}}_{t}, \bar{u}) + \frac{1}{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2}) + M_{1} (t) ‖ \nabla \bar{u} ‖^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} - α_{0} ‖ \bar{η} ‖_{M_{2}}^{2} \\ (3.15) & ⩽ L ‖ {\bar{u}}_{t} ‖^{2} + (\frac{1}{2} + ϵ) ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + C ‖ \bar{u} ‖^{2} . \end{aligned}$

(3.13) $+ ϵ \times$ (3.15) gives $\begin{aligned} (3.16) & \frac{d}{d t} H (t) + K (t) ⩽ C (‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖) ‖ \nabla \bar{u} ‖^{2} + C ‖ \bar{u} ‖^{2}, \end{aligned}$ where $\begin{aligned} H (t) = & \frac{1}{2} (ε (t) ‖ {\bar{u}}_{t} ‖^{2} + ‖ \nabla \bar{u} ‖^{2} + ‖ \bar{η} ‖_{M_{2}}^{2} + M_{1} (t) ‖ \nabla \bar{u} ‖^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2}) \\ + ϵ (ε (t) {\bar{u}}_{t}, \bar{u}) + \frac{ϵ}{2} ‖ \bar{u} ‖_{H_{0}^{1}}^{2} \\ \sim & ε (t) ‖ {\bar{u}}_{t} ‖^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + ‖ \bar{η} ‖_{M_{2}}^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2}, \\ K (t) = & (1 - ϵ - ϵ L) ‖ {\bar{u}}_{t} ‖^{2} + (\frac{1}{2} ϵ - 2 ϵ^{2}) ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + (\frac{δ_{2}}{2} - α_{0} ϵ) ‖ η ‖_{M_{2}}^{2} \\ + ϵ M_{1} (t) ‖ \nabla \bar{u} ‖^{2} + ϵ {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2} \\ \sim & ε (t) ‖ {\bar{u}}_{t} ‖_{H_{0}^{1}}^{2} + ‖ \bar{u} ‖_{H_{0}^{1}}^{2} + ‖ \bar{η} ‖_{M_{2}}^{2} + {\tilde{M}}_{1} (t) {(\nabla (u^{1} + u^{2}), \nabla \bar{u})}^{2} \end{aligned}$ for $ϵ > 0$ suitably small. Therefore, there exists a $κ > 0$ such that $\begin{aligned} (3.17) & K (t) - κ H (t) ⩾ 0 . \end{aligned}$

Inserting (3.17) into (3.16), we deduce $\begin{aligned} (3.18) & \frac{d}{d t} H (t) + κ H (t) ⩽ C (‖ \nabla u_{t}^{1} ‖ + ‖ \nabla u_{t}^{2} ‖) H (t) + C ‖ \bar{u} ‖^{2} . \end{aligned}$

By using the Gronwall inequality, we obtain $\begin{aligned} {‖ z_{1} (t) - z_{2} (t) ‖}_{H_{t}}^{2} ⩽ & C e^{- κ t} {‖ z_{1} (0) - z_{2} (0) ‖}_{H_{t}}^{2} \\ + C \int_{0}^{t} e^{- κ (t - τ)} ({‖ u^{1} (τ) - u^{2} (τ) ‖}^{2} d τ . \end{aligned}$

Then the proof of Lemma 3.2 is complete. □

4. Fractal dimension of the global attractor

Lemma 4.1 ([2]).

Let $(H, S (t))$ be given by ( 2.6 ) and it satisfies ( 2.7 ). If $(H, S (t))$ possesses a compact global attractor $A$ and is quasi-stable on $A$ , then the attractor $A$ has finite fractal dimension.

We now show that under assumptions (H1)–(H3), the inequality (3.8) implies (2.9), then our system is quasi-stable on the attractor $A$ .

Lemma 4.2.

Suppose the assumptions (H1)–(H3), ( 2.10 ) hold, then $(H, S (t))$ is quasi-stable on any bounded positively invariant set $B \subset H$ .

Proof.

Since $(H_{t}, S (t))$ is defined as the solution operator of (1.11)–(1.14), we can deduce (2.6) and (2.7) hold with $X = V_{2}$ , $Y = V_{1}$ and $Z = M_{2}$ . We see that condition (2.8) holds true, then we only need to verify stabilizability inequality (2.9).

Let $B \subset H$ be a bounded positively invariant with respect to $S (t)$ . For $z_{1}, z_{2} \in B$ , we write $S (t) z_{i} = (u^{i} (t), {(u^{i})}_{t} (t), η_{i}^{t})$ , $i = 1, 2$ . Define the seminorm $\begin{aligned} (4.1) & n_{X} (u) = ‖ u ‖^{2} . \end{aligned}$

We deduce that embeddings $H_{0}^{1} ↪ L^{2}$ is compact and $n_{X} (\cdot)$ is a compact seminorm on X. From (3.8), we derive $\begin{aligned} (4.2) & {‖ S (t) z_{1} - S (t) z_{2} ‖}_{H}^{2} ⩽ b (t) ‖ z_{1} - z_{2} ‖_{H}^{2} + c (t) sup_{0 < s < t} {[n_{X} (u^{1} (s) - u^{2} (s))]}^{2}, \end{aligned}$ where $\begin{aligned} (4.3) & b (t) = C e^{- κ t} and c (t) = C \int_{0}^{t} e^{- κ (t - τ)} d τ, t ⩾ 0 . \end{aligned}$

Moreover, we denote $b \in L^{1} (R^{+})$ and ${lim}_{t \to \infty} b (t) = 0$ . Also, since B is bounded it follows that $c (t)$ is locally bounded on $[0, \infty)$ . Consequently, $(H_{t}, S (t))$ is quasi-stable on any bounded positively invariant set $B \subset H$ . □

Theorem 4.3.

The global attractor $A$ in Theorem 2.5 has finite fractal dimension.

Proof.

From Lemma 4.2 and Theorem 4.1, this conclusion is obviously valid. □

Remark 4.1.

The Hausdorff dimension of $A$ is also finite, because the fractal dimension is always bigger than the Hausdorff dimension.

Footnotes

Acknowledgements

This work was supported by the NNSF of China with contract No. 12171082, the fundamental research funds for the central universities with contract numbers 2232022G-13, 2232023G-13 and by a grant from Science and Technology Commission of Shanghai Municipality. We are grateful to reviewers for their good suggestions and great help.

References

Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Diff. Equ. 252 (2012), 1229–1262. doi:10.1016/j.jde.2011.08.022.

Chueshov and

Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010.

De Brito, The damped elastic stretched string equation generalized: Existence, uniqueness, regularity and stability, Appl. Anal. 13(3) (1982), 219–233. doi:10.1080/00036818208839392.

Ding and

Yang, Attractors of the strongly damped Kirchhoff wave equation on

R^{N}

, Pure. Appl. Anal 18(2) (2019).

Ghisi and

Gobbino, Global existence and asymptotic behavior for a mildly degenerate dissipative hyperbolic equation of Kirchhoff type, Asymp. Anal. 40(1) (2004), 25–36.

Giorgi,

Muñoz Rivera and

Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl. 260(1) (2001), 83–99. doi:10.1006/jmaa.2001.7437.

Kirchhoff, Vorlesungen über mathematische Physik, 1883.

Li,

Yang and

Ding, Regular solutions and strong attractors for the Kirchhoff wave model with structural nonlinear damping, Appl. Math. Letters 104 (2020), 106258. doi:10.1016/j.aml.2020.106258.

Lin,

Lv and

Lou, Exponential attractors and inertial manifolds for a class of nonlinear generalized Kirchhoff-Boussinesq model, Far East J. Math. Sci. 101(9) (2017), 1913–1945.

10.

Lv,

Lin and

Lv, Well-posedness and global attractor of Kirchhoff equation with memory term and thermal effect, Results Appl. Math. 18 (2003), 100362. doi:10.1016/j.rinam.2023.100362.

11.

Ma,

Zhang and

Zhong, Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping, Disc. Cont. Dynam. Syst. Series B 24(9) (2019).

12.

Ma,

Zhang and

Zhong, Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension, J. Math. Anal. Appl. 484 (2020), 123670. doi:10.1016/j.jmaa.2019.123670.

13.

Nishihara and

Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcial. Ekvac. 33 (1990), 151–159.

14.

M.J.

Silva and

Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys. 54 (2013), 021505. doi:10.1063/1.4792606.

15.

Solange and

Koumou Patcheu, Global existence and exponential decay estimates for damped quasilinear equation, Comm. Partial Diff. Equ. 22(11–12) (1997), 2007–2024. doi:10.1080/03605309708821328.

16.

Yamada, On some quasilinear wave equations with dissipative terms, Nagoya Math. J. 87 (1982), 17–39. doi:10.1017/S0027763000019929.

17.

Yang,

Qin,

Miranville and

Wang, Existence and regularity of global attractors for a Kirchhoff wave equation with strong damping and memory, 2023, submitted.

18.

Yang, Longtime behavior of the Kirchhoff type equation with strong damping on

R^{N}

, J. Diff. Equ. 242 (2007), 269–286. doi:10.1016/j.jde.2007.08.004.

19.

Yang,

Ding and

Liu, Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity, Appl. Math. Letters 33 (2014), 12–17. doi:10.1016/j.aml.2014.02.014.

20.

Yang and

Li, Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations, Disc. Cont. Dynam. Syst. Series B 24(9) (2019), 4899–4912.

21.

Yang and

Liu, Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity, Appl. Math. Letters 46 (2015), 127–132. doi:10.1016/j.aml.2015.02.019.