Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

TL;DR

This paper proves the global stability of travelling front solutions in a damped wave equation with bistable potential, showing solutions converge to these fronts under certain initial conditions.

Contribution

It establishes the global stability of travelling fronts for a damped wave equation with bistable nonlinearity, extending previous results to this specific system.

Findings

Solutions converge uniformly to travelling fronts over time

Stability holds for initial data close to the front profile

Lyapunov function in Galilean frame is key to the proof

Abstract

We consider the damped wave equation \alpha u_tt + u_t = u_xx - V'(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x,t) = h(x-st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t goes to plus infinity. The proof of this global stability result is inspired by a recent work of E. Risler and relies on the fact that our system has a Lyapunov function in any Galilean frame.