A geometric analysis of front propagation in a family of degenerate reaction-diffusion equations with cutoff

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Abstract

We investigate the effects of a Heaviside cutoff on the dynamics of traveling fronts in a family of scalar reaction-diffusion equations with degenerate polynomial potential that includes the classical Zeldovich equation. We prove the existence and uniqueness of front solutions in the presence of the cutoff, and we derive the leading-order asymptotics of the corresponding propagation speed in terms of the cutoff parameter. For the Zeldovich equation, an explicit solution to the equation without cutoff is known, which allows us to calculate higher-order terms in the resulting expansion for the front speed; in particular, we prove the occurrence of logarithmic (switchback) terms in that case. Our analysis relies on geometric methods from dynamical systems theory and, in particular, on the desingularization technique known as 'blow-up.'.

Original languageEnglish
Pages (from-to)405-437
Number of pages33
JournalZeitschrift für angewandte Mathematik und Physik
Volume62
Issue number3
DOIs
Publication statusPublished - Jun 2011

Keywords

  • Reaction-diffusion
  • Front propagation
  • Cutoff
  • Front speed
  • Geometric desingularization
  • Asymptotics
  • RIGOROUS ASYMPTOTIC EXPANSIONS
  • CRITICAL WAVE SPEEDS
  • BLOW-UP

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