Attractors for strongly damped wave equations with critical nonlinearities. (Q1879991)

The authors consider the following strongly damped wave equation in a bounded smooth domain \(\Omega\) of \(\mathbb R^n\): \[ \begin{aligned} u_{tt}+\eta(-\Delta)^\theta u_t-\Delta u=f(u),&\quad x\in\Omega,\\ u(0,x)=u_0(x),\;\;u_t(0,x)=v_0(x),&\quad x\in\Omega,\tag{1}\\ u(t,x)=0, &\quad x\in\partial\Omega, \end{aligned} \] under the assumptions that \(\theta\in[1/2,1]\) and \(f\) is dissipative and satisfies the following growth restriction: \[ | f(u_1)-f(u_2)| \leq c| u_1-u_2| (1+| u_1| ^{\rho-1}+| u_2| ^{\rho-1}),\;\;u_1,u_2\in\mathbb R \tag{2} \] with \(\rho\leq\frac{n+2}{n-2}\). The main result of the paper is the existence of a compact global attractor (in the standard energy phase space) in the following cases: 1) \(\theta\in[1/2,1)\), \(\rho<\frac{n+2}{n-2}\). 2) \(\theta=1\), \(\rho\leq\frac{n+2}{n-2}\). The proof is strongly based on the fact that the linear part of (1) generates an analytic semigroup and on the previous results of the authors on the local well-posedness of (1) [see the authors, Bull. Aust. Math. Soc. 66, 443--463 (2002; Zbl 1020.35059)].