Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure

Abstract

We consider nonlinear parabolic systems of the form $u_t=-\nabla V(u)+u_{xx}$, where $u\in {\mathbb{R}}^n$, $n⩾1$, $x\in \mathbb{R}$, and the potential V is coercive at infinity. For such systems, we prove a result of global convergence toward bistable fronts which states that invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via a traveling front connecting to another (lower) equilibrium. This provides, for instance, a generalization of the global convergence result obtained by Fife and McLeod [P. Fife, J.B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335–361] in the case $n=1$. The proof is based purely on energy methods, it does not make use of comparison principles, which do not hold any more when $n>1$.