# Metastability for nonlinear convection–diffusion equations

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 $$\varepsilon$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Differential Equations and Applications NoDEA Springer Journals

# Metastability for nonlinear convection–diffusion equations

, Volume 24 (4) – Jun 13, 2017
20 pages

/lp/springer_journal/metastability-for-nonlinear-convection-diffusion-equations-EQcVvg6YB2
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
1021-9722
eISSN
1420-9004
D.O.I.
10.1007/s00030-017-0459-5
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Nonlinear Differential Equations and Applications NoDEASpringer Journals

Published: Jun 13, 2017

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