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: $1\leqslant p<p_\alpha$: (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 $\alpha_0\in (1/2, 1)$.