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Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Bin Yang, Yuming Qin, Alain Miranville, Ke Wang

Abstract

In this paper, we shall investigate the existence and upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with a strong damping in the time-dependent space $X_t$. After deriving the existence and uniqueness of solutions by the Faedo-Galerkin approximation method, we establish the existence of pullback attractors. Later on, we prove the upper semicontinuity of pullback attractors between the Kirchhoff-type wave equations with $δ\geq 0$ and the conventional wave equations with $δ=0$ by a series of complex energy estimates.

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

Abstract

In this paper, we shall investigate the existence and upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with a strong damping in the time-dependent space . After deriving the existence and uniqueness of solutions by the Faedo-Galerkin approximation method, we establish the existence of pullback attractors. Later on, we prove the upper semicontinuity of pullback attractors between the Kirchhoff-type wave equations with and the conventional wave equations with by a series of complex energy estimates.
Paper Structure (4 sections, 16 theorems, 177 equations)

This paper contains 4 sections, 16 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of pullback attractors $\mathcal{A}_\delta=\{A_\delta(t)\}_{t\in \mathbb R}$
  4. Upper semicontinuity of pullback attractors

Key Result

Lemma 2.6

(clr.6) Suppose the family $\mathcal{D}=\{D(t)\}_{t \in \mathbb{R}}$ is pullback absorbing and the process $U(\cdot, \cdot)$ is pullback $\mathcal{D}$-asymptotically compact in $X$, if the family $\mathcal{A}=\{A(t)\}_{t \in \mathbb{R}}$ satisfies then $\mathcal{A}$ is a pullback attractor for $U(\cdot, \cdot)$ in $X$. In addition, if for any $t \in \mathbb{R}$, there exists a $T=T(t)>0$ such tha

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 13 more