Comparison theorems and variable speed waves for a scalar reaction-diffusion equation

Alison L. Kay, Jonathan A. Sherratt, J. B. McLeod

Research output: Contribution to journalLiterature review

14 Citations (Scopus)

Abstract

This paper concerns the reaction-diffusion equation ut = uxx + u2(1 - u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed. © 2001 The Royal Society of Edinburgh.

Original languageEnglish
Pages (from-to)1133-1161
Number of pages29
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume131
Issue number5
Publication statusPublished - 2001

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