On Transitions to Stationary States in One‐Dimensional Nonlinear Wave Equations

Abstract

. We consider the long‐time asymptotics of solutions to one‐dimensional nonlinear wave equations, which are infinite‐dimensional Hamiltonian systems. We assume that the nonlinear term is concentrated at a finite segment of the line. We prove long‐time convergence to stationary states for all finite‐energy solutions in the Fréchet topology defined by local energy seminorms. This means that the set of stationary states is a point attractor for the systems in the Fréchet topology. The investigation is inspired by N. Bohr's postulate on the transitions between stationary states in quantum systems.