Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

J. C. Meyer, D. J. Needham*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
124 Downloads (Pure)

Abstract

In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate apbq (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present.
Original languageEnglish
Article number20140632
JournalRoyal Society of London. Proceedings A. Mathematical, Physical and Engineering Sciences
Volume471
Issue number2175
Early online date21 Jan 2015
DOIs
Publication statusPublished - 1 Mar 2015

Keywords

  • semi-linear parabolic partial differential equation
  • non-Lipschitz nonlinearity
  • well-posed

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy

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