A hyperbolic variant of Simon's convergence theorem (Q2708569)

This is a survey of results achieved by the author and his collaborators on the long-time behaviour of solutions to the semilinear damped wave equation of the form: NEWLINE\[NEWLINE u_{tt} + c u_t - \Delta u + f(x,u) = 0,NEWLINE\]NEWLINE posed on a bounded spatial domain and complemented by the homogeneous Dirichlet boundary conditions. It is shown that any global-in-time bounded solution converges as time tends to infinity to a single equilibrium provided the nonlinearity \(f\) is analytic with respect to the \(u\) variable. A refined technique based on the fundamental result of Lojasiewicz is used to prove the result.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].