Metastability for nonlinear convection–diffusion equations

We study the metastable dynamics of solutions to nonlinear evolutive equations of parabolic type, with a particular attention to the case of the viscous scalar Burgers equation with small viscosity ε. In order to describe rigorously such slow motion, we adapt the strategy firstly proposed in Mascia and Strani (SIAM J Math Anal 45:3084–3113, 2013) by linearizing the original equation around a metastable state and by studying the system obtained for the couple (ξ, v), where ξ is the position of the internal shock layer and v is a perturbative term. The main result of this paper provides estimates for the speed of the shock layer and for the error v; in particular, in the case of the viscous Burgers equation, we prove they are exponentially small in ε. As a consequence, the time taken for the solution to reach the unique stable steady state is exponentially large, and we have exponentially slow motion.