Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data

Abstract

Consider the following equations: \({(E)\ \ u_t-\Delta u=u^p}$, ${(E')\ \ u_t-\Delta u=u^p-\mu\mid\nabla u\mid^q}$, ${(E'')\ \ u_t-\Delta u=u^p+a.\nabla (u^q)}$, in ${\Omega\subset I\!\!R^d}\). For any unbounded domain \(\Omega\), intermediate between a cone and a strip, we obtain a sufficient condition on the decay at infinity of initial data to have blow-up. This condition is related to the geometric nature of \({\Omega}\). For instance, if \(\Omega \) is the interior of a revolution surface of the form \({\mid x'_d\mid f(\mid x_d\mid )}\), then the condition on the initial data is given by \({\Phi (x)>Cf(\mid x\mid )^{-2/(p-1)}}\) at infinity. Moreover, for a large class of domains \({\Omega}\), we prove that those results are optimal (i.e. there exist global solutions with the same order of decay at infinity for their initial data).