COMPLETE BLOW-UP AND THERMAL AVALANCHE FOR HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

Abstract

ABSTRACT We study the continuation after blow-up of solutions u(x,t) of the heat equation, with a nonlinear flux condition at the boundary, . We prove that such a continuation is trivial (i.e., identically infinite for all t>T, where T is the blow-up time) in the following sense: if we replace f(u) by a sequence of functions fn (u) that do not cause blow-up and fn converges to f uniformly on compact sets, then the corresponding solutions un satisfy for every x > 0 and every t>T. This is called complete blow-up and happens in the present problem for all positive and continuous functions f for which solutions blow up. An interesting phenomenon related to complete blow-up is the thermal avalanche: in the cases in which there is single-point blow-up as the singularity at the origin propagates instantaneously at time t = T to cover the whole space. We describe the formation of the avalanche in the case , as a boundary layer which appears in the limit of the approximate problems by choosing a suitable scaling and passing to self-similar variables. We then show that the layer is described by the solution of a limit problem. We also describe the asymptotic behaviour for the approximate problems as t goes to infinity. We also consider the nonlinear diffusion equation , with nonlinear boundary condition . We prove that blow-up solutions have a nontrivial continuation if and only if m < 1, independently of f. The thermal avalanche when m > 1 and f is a power is also described in this case, as well as the asymptotic behaviour for the approximations for all m.