Global attractors and their upper semicontinuity for a structural damped wave equation with supercritical nonlinearity on ${\mathbb{R}}^N$

Abstract

The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on ${\mathbb{R}}^N:u_{tt}-\Delta u+(-\Delta {)}^{\alpha }{u}_t+u_t+u+g(u)=f(x)$, where $\alpha \in (1/2,1)$ is called a dissipative index. We propose a new method based on the harmonic analysis technique and the commutator estimate to exploit the dissipative effect of the structural damping $(-\Delta {)}^{\alpha }{u}_t$ and to overcome the essential difficulty: "both the unbounded domain ${\mathbb{R}}^N$ and the supercritical nonlinearity cause that the Sobolev embedding loses its compactness"; Meanwhile we show that there exists a supercritical index $p_{\alpha }\equiv \frac{N+4\alpha }{N-4\alpha }$ depending on $\alpha$ such that when the growth exponent $p$ of the nonlinearity $g(u)$ is up to the supercritical range: $1⩽p<p_{\alpha }$: (i) the IVP of the equation is well-posed and its solution is of additionally global smoothness when $t>0$; (ii) the related solution semigroup possesses a global attractor ${\mathcal{A}}_{\alpha }$ in natural energy space for each $\alpha \in (1/2,1)$; (iii) the family of global attractors $\{{\mathcal{A}}_{\alpha }{\}}_{\alpha \in (1/2,1)}$ is upper semicontinuous at each point ${\alpha }_0\in (1/2,1)$.

23 pages