Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws

The aim of this paper is to contribute to the definition of a versatile language for metastability in the context of partial differential equations of evolutive type. A general framework suited for parabolic equations in one-dimensional bounded domains is proposed, based on choosing a family of approximate steady states Uε(ε)εJ and on the spectral properties of the linearized operators at such states. The slow motion for solutions belonging to a cylindrical neighborhood of the family Uε is analyzed by means of a system of an ODE for the parameter ε = ε(t), coupled with a PDE describing the evolution of the perturbation v := u - Uε(ε). We state and prove a general result concerning the reduced system for the couple (ε, v), called quasi-linearized system, obtained by disregarding the nonlinear term in v, and we show how such an approach suits to the prototypical example of scalar viscous conservation laws with Dirichlet boundary conditions in a bounded one-dimensional interval with convex flux. © 2013 Society for Industrial and Applied Mathematics