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

Emmanuel Risler

Published 2016-04-07Version 1

This paper is concerned with spatially extended gradient systems of the form \[ u_t=-\nabla V (u) + u_{xx}\,, \] where spatial domain is the whole real line, state-parameter $u$ is multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions of the potential, the asymptotic behaviour of every \emph{bistable solution} --- that is, every solution close at both ends of space to spatially homogeneous stable equilibria --- is described. Every such solutions approaches, far to the left in space a stacked combination of bistable fronts travelling to the left, far to the right in space a stacked combination of bistable fronts travelling to the right, and in between a pattern of stationary solutions homoclinic or heteroclinic to homogeneous equilibria. This result pushes one step further the program initiated in the late seventies by Fife and MacLeod about the global asymptotic behaviour of bistable solutions, by extending their results to the case of systems. In the absence of maximum principle, the arguments are purely variational, and call upon previous results obtained in companion papers.