Global behaviour of bistable solutions for hyperbolic gradient systems in one unbounded spatial dimension

TL;DR

This paper analyzes the long-term behavior of bistable solutions in damped hyperbolic gradient systems on an unbounded domain, revealing their convergence to traveling fronts and stationary profiles under generic conditions.

Contribution

It extends previous results to hyperbolic systems without maximum principle, describing the global asymptotic structure of bistable solutions using purely variational methods.

Findings

Solutions approach traveling fronts at spatial infinities.

Stationary profiles form between traveling fronts.

Results generalize previous parabolic and wave equation analyses.

Abstract

This paper is concerned with damped hyperbolic gradient systems of the form \[ \alpha u_{tt} + u_t = -\nabla V(u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\alpha$ is a positive quantity, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of every bistable solution (that is, every solution close at both ends of space to stable homogeneous equilibria) is described. Every such solution approaches, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right in space a stacked family of bistable fronts travelling to the right, and in between a pattern of profiles of stationary solutions homoclinic or heteroclinic to stable homogeneous equilibria, going slowly away from one another. In the absence of maximum principle, the arguments are purely variational. This extends previous results obtained in companion papers for damped wave equations or parabolic gradient systems, in the spirit of the program initiated in the late seventies by Fife and McLeod on the global asymptotic behaviour of bistable solutions for parabolic equations.