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 $$\varepsilon $$ 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 $$(\xi ,v)$$ ( ξ , v ) , where $$\xi $$ 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 $$\varepsilon $$ 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.
Nonlinear Differential Equations and Applications NoDEA – Springer Journals
Published: Jun 13, 2017
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