Stabilization of solutions of the diffusion equation with a non-lipschitz reaction term

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Abstract

In this paper we are concerned with the reaction-diffusion equation ut = Δu + f(u) in a ball of RN with Dirichlet boundary condition. We assume that f satisfies the concave-convex condition. A typical example is f(u) = |u|q-1u + |u|p-1u (0 < q < 1 < p < (N+2)/(N-2)). First we obtain the complete structure of positive solutions to the stationary problem; Δφ + f(φ) = 0. Next we state the relations between this structure and time-depending behaviors of nonnegative solutions (global existence or blow up) to the non-stationary problem.

Original languageEnglish
Pages (from-to)789-800
Number of pages12
JournalNonlinear Analysis, Theory, Methods and Applications
Volume47
Issue number2
DOIs
Publication statusPublished - 2001 Aug 1
Event3rd World Congres of Nonlinear Analysts - Catania, Sicily, Italy
Duration: 2000 Jul 192000 Jul 26

Keywords

  • Blow up
  • Comparison theorem
  • Global solution
  • Non-Lipschitzian nonlinearity
  • Radially symmetric solution
  • Reaction-diffusion equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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