Isolated singularities and asymptotic behaviour of the solutions of a semi-linear heat equation

Abstract

We study here the limit properties of the solutions of the following singular problem (1) u_t−Δu+u|u|^p−l=0 in R^N×R⁺\{(0,0)} with u(x,0)=0 if x≠(0,0). We transform the equation from an autonomous dynamical system into a weighted Sobolev space and look for the limit set of a trajectory of such a system. By using an energy method, we obtain a theorem of classification of singularities of the solutions of (1). By using the same methods we study the asymptotic behaviour of the solutions and some global singular solutions.