Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions.

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction-diffusion equation ut−Δu=f(x,u) in a smooth domain Ω with nonlinear boundary conditions ∂u∂n=g(x,u). We show that, if locally around some point of the boundary, we have f(x,u)=−βup,β≥0, and g(x,u)=uq, then blow-up in finite time occurs if 2q>p+1 or if 2q=p+1 and β<q. Moreover, if we denote by Tb the blow-up time, we show that a proper continuation of the blow-up solutions are pinned to the value infinity for some time interval [T,τ] with Tb≤T<τ. On the other hand, for the case f(x,u)=−βup, for all x and u, with β>0 and p>1, we show that blow-up occurs only on the boundary.