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

Emmanuel Risler

Published 2016-04-04Version 1

This paper is concerned with spatially extended gradient systems of the form \[ u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,, \] where spatial domain is the whole real line, state-parameter $u$ is multidimensional, $\mathcal{D}$ denotes a fixed diffusion matrix, and the potential $V$ is coercive at infinity. \emph{Bistable} solutions, that is solutions close at both ends of space to stable homogeneous equilibria, are considered. For a solution of this kind, it is proved that, if the homogeneous equilibria approached at both ends belong to the same level set of the potential and if an appropriate (localized in space) energy remains bounded from below when time increases, then the solution approaches, when time approaches infinity, a pattern of stationary solutions homoclinic or heteroclinic to homogeneous equilibria. This result provides a step towards a complete description of the global behaviour of all bistable solutions that is pursued in a companion paper. Some consequences are derived, and applications to some examples are given.