A very singular solution of the porous media equation with absorption

Abstract

turns out to play an important role in the description of both the short- and the long-time behaviour of positive solutions of Eq. (1.1). For the short-time behaviour it provides, together with the one- parameter family of singular solutions I’,. of Problem (I), a complete description of the possible types of singular behaviour at the point (0,O) [ 121. For the long-time behaviour of solutions u(x, t) of the Cauchy Problem for (1.1 ), (1.2) it provides the asymptotic form when the initial function decays sufficiently fast as 1x1 + co: t”(P- ‘){u(x, t) - W(x, t,} + 0, ast+co uniformly in R” [7, 81. In this paper we shall show that if