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

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.