Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

Honglv Ma,Jun Xie,Jing Wang,Sergey Shmarev

doi:10.1155/2021/7575078

Honglv Ma, Jun Xie + Show 2 more

Open Access

https://doi.org/10.1155/2021/7575078

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Abstract

In this paper, we obtain the existence of pullback attractors for nonautonomous Kirchhoff equations with strong damping, which covers the case of possible generation of the stiffness coefficient. For this purpose, a necessary method via “the measure of noncompactness” is established.

Highlights

Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω
We call the Kirchhoff equation nondegenerate if the stiffness φ satisfies the strict hyperbolicity condition φðsÞ ≥ c > 0 and degenerate if φðsÞ ≥ 0 on R+
Concerning Kirchhoff equations with strong dissipation, the first result on the well-posedness we are aware of was obtained by Nishihara [16]

Summary

Introduction

Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω. We consider the following Kirchhoff wave model with strong damping: utt − Δut − φÀk∇uk2ÁΔu + f ðuÞ = hðx, tÞ, in Ω × ðτ,∞Þ, uj∂Ω = 0, uðx, τÞ = u0τðxÞ, utðx, τÞ = u1τðxÞ, x ∈ Ω, τ ∈ R, ð1Þ where hðx, tÞ is a time-dependent external force term, u0τ and u1τ are initial data, and φ and f are nonlinear functions specified later. Concerning Kirchhoff equations with strong dissipation, the first result on the well-posedness we are aware of was obtained by Nishihara [16] He proved the global existence of the solution for the model utt − Δut − mðk∇ukÞΔu = 0. In the case of possible degeneration of the stiffness coefficient and the case of Advances in Mathematical Physics supercritical source term (p∗ < p < ðn + 4Þ/ðn − 4Þ+), the first result on the well-posedness we are aware of is given by Chueshov [25] When he proved the existence of a global attractor for problem (1) in the natural energy space ðH ðΩÞ.

Preliminaries

Existence of Pullback Attractors