Asymptotic behavior for a semilinear second order evolution equation

Abstract

This paper is devoted to the qualitative analysis for a second order evolution equation $u_{tt}-\Delta u-\Delta u_t -\varepsilon \Delta u_{tt}+ f(u)=g(x)$ $(\varepsilon \in [0,1])$ with critical nonlinearity. Some uniformly (w.r.t. $\varepsilon \in [0,1]$) asymptotic regularity about the solutions has been established for both $g(x)\in L^2(\Omega )$ and $g(x)\in H^{-1}$, which shows that the solutions are exponentially approaching a more regular fixed subset uniformly (w.r.t. $\varepsilon \in [0,1]$). As an application of this regularity result, a family $\{\mathcal {E}_{\varepsilon }\}_{\varepsilon \in [0,1]}$ of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with a strongly damped wave equation ($\varepsilon =0$), the upper semicontinuity, at $\varepsilon =0$, of the global attractors has been proved.