Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models

The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: $ u_{tt}-(1+\epsilon\|\nabla u\|^{2})\Delta u-\Delta u_{t}+f(u) = g(x,t) $, where $ \epsilon\in [0,1] $ is an extensibility parameter. It shows that when the nonlinearity $ f(u) $ is of optimal supercritical growth $ p: \frac{N+2}{N-2} = p^*<p<p^{**} = \frac{N+4}{(N-4)^+} $: (ⅰ) the related evolution process has in natural energy space $ \mathcal{H} = (H^1_0\cap L^{p+1})\times L^2 $ a compact uniform attractor $ \mathcal{A}^{\epsilon}_{\Sigma} $ for each $ \epsilon\in [0,1] $; (ⅱ) the family of compact uniform attractor $ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $ is continuous on $ \epsilon $ in a residual set $ I^*\subset [0,1] $ in the sense of $ \mathcal{H}_{ps} ( = (H^1_0\cap L^{p+1,w})\times L^2) $-topology; (ⅲ) $ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $ is upper semicontinuous on $ \epsilon\in [0,1] $ in $ \mathcal{H}_{ps} $-topology.