Transient behavior of solutions to a class of nonlinear boundary value problems

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Abstract

In this paper we consider the asymptotic behavior in time of solutions to the heat equation with nonlinear Neumann boundary conditions of the form ∂u/∂n = F(u), where F is a function that grows superlinearly. Solutions frequently exist for only a finite time before "blowing up." In particular, it is well known that solutions with initial data of one sign must blow up in finite time, but the situation for sign-changing initial data is less well understood. We examine in detail conditions under which solutions with sign-changing initial data (and certain symmetries) must blow up, and also conditions under which solutions actually decay to zero. We carry out this analysis in one space dimension for a rather general F, while in two space dimensions we confine our analysis to the unit disk and F of a special form.

Original languageEnglish (US)
Pages (from-to)261-290
Number of pages30
JournalQuarterly of Applied Mathematics
Volume69
Issue number2
DOIs
StatePublished - 2011

All Science Journal Classification (ASJC) codes

  • Applied Mathematics

Keywords

  • Blowup
  • Heat equation
  • Nonlinear Neumann boundary condition

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