Traveling waves of dissipative nonautonomous hyperbolic equations in a strip (Q1977675)

The longtime behavior of solutions of the nonlinear damped wave equation \[ \rho u_{tt}+\eta u_t-\Delta_x u+f(u)=g_0(t,x),\;\rho,\eta>0,\;\;x\in\Omega\subseteq\mathbb{R}^n \tag{1} \] with the nonautonomous external force \(g_0(t,x)\) (which is assumed to be translation-compact with respect to \(t\)) is studied. It is proved, that under natural dissipativity and growth restrictions on the nonlinear interaction function \(f\) this longtime behavior can be described in terms of the attractor of the semigroup of temporal shifts acting on the trajectory phase space \(K^+\) which is a collection of all solutions \(\{u(t,x),t\geq 0\}\) for the family of equations (1) with all external forces \(g\) belonging to the hull of the initial external force \(g_0\) [see also \textit{V. Chepyzhov} and \textit{M. I. Vishik}, J. Math. Pures Appl., IX. Ser. 76, No 10, 913-964 (1997; Zbl 0896.35032)]. The relations between the obtained attractor of (1) and fast (supersonic) travelling waves in the corresponding hyperbolic equation in the unbounded cylindrical domain \(\mathbb{R}\times\Omega\) are also indicated.