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

Bryan, Kurt; Vogelius, Michael

doi:10.1090/S0033-569X-2011-01204-5

Authors: Kurt Bryan and Michael S. Vogelius
Journal: Quart. Appl. Math. 69 (2011), 261-290
MSC (2000): Primary 35B05, 35B40
DOI: https://doi.org/10.1090/S0033-569X-2011-01204-5
Published electronically: March 3, 2011
MathSciNet review: 2814527
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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 $\partial u/\partial \mathbf {n}=F(u)$, 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.

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