Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

J. C. Meyer, D. J. Needham

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Abstract

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

Original languageEnglish
Pages (from-to)777-832
JournalProceedings of the Royal Society of Edinburgh: Section A (Mathematics)
Volume146
Issue number4
Early online date19 Jul 2016
DOIs
Publication statusPublished - 1 Aug 2016

Keywords

  • Hadamard well-posedness
  • non-Lipschitz nonlinearity
  • parabolic semilinear partial differential equations

ASJC Scopus subject areas

  • Mathematics(all)

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