Bounded, locally compact global attractors for semilinear damped wave equations on \(\mathbb{R}^ N\) (Q1916343)

The impact of dissipation on the long-time behavior of the nonlinear waves governed by the model equation \[ u_{tt}+u_t-\Delta u+f(u)=0, \quad u=u(x,t), \qquad x\in\mathbb{R}^N,\tag{1} \] is examined, where \(f\in C^1(\mathbb{R}^1)\), \[ \lim_{|z|\to\infty}\inf {{f(z)}\over z}\geq a>0,\quad f'(z)\geq-C\qquad \text{for all }z\in\mathbb{R}^1. \] The author shows the existence of a global attractor for the equation (1) and that the attractor is compact in \(H^1_{\text{loc}} (\mathbb{R}^N)\times L^2_{\text{loc}} (\mathbb{R}^N)\).