Abstract

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			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. View/Download Publication in pdf format Back to Preprint list Michael Vogelius 2-13-98