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 partial derivative u/partial derivative 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.