Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction-diffusion equations

Peter De Maesschalck, Nikola Popovic

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Abstract

We consider front propagation in a family of scalar reaction-diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity. We investigate the Gevrey properties of the corresponding critical propagation speed, proving that the formal series expansion for that speed is Gevrey-1 with respect to the inverse of the degree. Moreover, we discuss the question of optimal truncation. Finally, we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order, and we illustrate that algorithm by calculating explicitly the first ten coefficients. Our analysis builds on results obtained previously in [F. Dumortier, N. Popovic, T.J. Kaper, The asymptotic critical wave speed in a family of scalar reaction-diffusion equations, J. Math. Anal. Appl. 326 (2) (2007) 1007-1023], and makes use of the blow-up technique in combination with geometric singular perturbation theory and complex analysis, while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic. (C) 2011 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)542-558
Number of pages17
JournalJournal of mathematical analysis and applications
Volume386
Issue number2
DOIs
Publication statusPublished - 15 Feb 2012

Keywords

  • Reaction-diffusion equations
  • Front propagation
  • Critical wave speeds
  • Asymptotic expansions
  • Blow-up technique
  • Gevrey asymptotics
  • Optimal truncation
  • SINGULAR PERTURBATION-THEORY
  • DIFFERENTIAL-EQUATIONS

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