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The Influence of Gradient Perturbations on Blow-up Asymptotics in Semilinear Parabolic Problems: A Survey

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Nonlinear Elliptic and Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 64))

Abstract

In this paper, we give a survey of results concerning semilinear parabolic problems of the form u t - Δu = u p + g(x, t, u, ∇u). Our goal is to examine the effect of the (gradient) perturbation term g on the asymptotic behavior of blow-up solutions. It turns out that, if the perturbation becomes critical or supercritical in a scaling sense, then the blow-up rate as well as the blow-up profiles may become notably different from those known in the unperturbed case. In some cases, we give precise asymptotic estimates on blow-up solutions.

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Author information

Authors and Affiliations

  1. Dëpartement de Mathëmatiques INSSET, Université de Picardie, F-02109, St-Quentin, France

    Philippe Souplet

  2. Laboratoire de Mathématiques Appliquées UMR CNRS 7641, Université de Versailles, 45 avenue des Etats-Unis, F-78035, Versailles, France

    Philippe Souplet

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  1. Philippe Souplet
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Editor information

Editors and Affiliations

  1. Universitá Pierre et Marie Curie, Paris

    Haim Brezis

  2. Rutgers University, New Brunswick, N.J.

    Haim Brezis

  3. Angewandte Mathematik, Universität Zurich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  4. Institute of Applied Mathematics, University of Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Souplet, P. (2005). The Influence of Gradient Perturbations on Blow-up Asymptotics in Semilinear Parabolic Problems: A Survey. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_28

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