Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium

Marcello Lucia, Cyrill B. Muratov, Matteo Novaga

Research output: Contribution to journalArticlepeer-review

44 Scopus citations

Abstract

We revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction-diffusion equations with one linearly stable and one linearly unstable equilibrium. For a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension. We introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism. We obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable. Our method also allows us to obtain efficient lower and upper bounds for the propagation speed.

Original languageEnglish (US)
Pages (from-to)616-636
Number of pages21
JournalCommunications on Pure and Applied Mathematics
Volume57
Issue number5
DOIs
StatePublished - May 2004

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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