Robustness of Attractors for Non-autonomous Kirchhoff Wave Models with Strong Nonlinear Damping

Abstract

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: $u_{tt}- (1+\epsilon \Vert \nabla u\Vert ^2)\Delta u-\sigma (\Vert \nabla u\Vert ^2) \Delta u_t+f(u)=g(x,t)$, where $\epsilon \in [0,1]$ is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: $1\le p<p^{**}\equiv \frac{N+4}{(N-4)^+}$, (i) the related process has a pullback attractor ${\mathscr {A}}_\epsilon$ in natural energy space ${\mathcal {H}}=(H^1_0\cap L^{p+1})\times L^2$ for each $\epsilon$, and it is upper semicontinuous on the perturbation $\epsilon$; (ii) the related process has a partially strong pullback exponential attractor ${\mathcal {M}}_\epsilon$ for each $\epsilon$, and it is Hölder continuous on $\epsilon \in [0,1]$. These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case.