Uniform attractors of non-autonomous Kirchhoff wave models

Abstract

The paper investigates the existence and upper semicontinuity of uniform attractors of the perturbed non-autonomous Kirchhoff wave equations with strong damping and supercritical nonlinearity: $u_{tt}-\Delta u_t-(1+\epsilon \Vert \nabla u{\Vert }^2)\Delta u+f(u)=g(x,t)$, where $\epsilon \in [0,1]$ is a perturbed parameter. It shows that when the nonlinearity $f(u)$ is of supercritical growth $p:\frac{N+2}{N-2}=p^{\ast }<p<p^{\ast \ast }=\frac{N+4}{(N-4{)}^{+}}$: (i) the related evolution process has a compact uniform attractor $\mathcal{A}_\ls^\e $ for each $\epsilon \in [0,1]$; (ii) the family of uniform attractor $\mathcal{A}_\ls^\e $ is upper semicontinuous on the perturbed parameter $\epsilon$ in the sense of partially strong topology.