Strong attractors and their continuity for the semilinear wave equations with fractional damping

Abstract

The paper investigates the existence of strong global and exponential attractors and their continuity for a class of semilinear wave equations with fractional damping in $\Omega\subset \mathbb R^3$: $$ u_{tt}-\Delta u+\gamma (-\Delta)^\theta u_t +f(u)=g(x), $$ with dissipative index $\theta \in (0, {\frac 12 } )$, It shows that when the growth exponent $p$ of the nonlinearity $f(u)$ is up to the range: $1\leq p < p_\theta:=3+4\theta$, (i) the weak solutions of the IBVP of the equation are of optimal higher regularity as $t > 0$, which leads to the fact that the weak solutions are exactly the strong ones; (ii) the related solution semigroup $S^\theta(t)$ acting on natural energy space ${\mathcal H}$ has a strong $({\mathcal H},{\mathcal H}_1)$-global attractor ${\mathscr A}_\theta$ and a strong $({\mathcal H},{\mathcal H}_1)$-exponential attractor ${\mathscr E}_\theta$, which are upper semicontinuous and continuous on $\theta$ at each point $\theta_0\in (0, {\frac 12 } )$ in the topology of strong space ${\mathcal H}_1$, respectively. The novelty of the paper is that it provides a new approach to establish the existence of the strong solutions while the initial data is in weaker space, and to study the existence of the strong global and exponential attractors and their continuity on the dissipative index $\theta$ in the topology of the strong space. These results improve and deepen those in recent literatures [35, 36, 37, 38].