Forward self-similar solutions of a semilinear parabolic equation with a nonlinear gradient term

Abstract

In this paper we consider the equation (E) $ \partial _{t} \Psi = \Delta \Psi + a \mid \nabla \Psi \mid ^{q} + \mid \Psi \mid^{p-1}\Psi ,$ $ t >0 , $ $ x\in \Bbb R^{n},$ which was first studied by Chipot and Weissler in a bounded domain with $a = -1 $. For $a$ sufficiently small, $q = \frac{2p }{ p+1 }$ the critical exponent and $1<{ n(p-1) \over 2 } < p+1,$ we establish the existence of a positive global self-similar solution of (E) with a singular initial data at the origin. In particular, this implies a nonuniqueness result for the Cauchy problem associated with (E) in $L^{s}( \Bbb R ^{n})$, where $ 1 \leq s < {n(p-1) \over {2}},$ and $p,$ $q$ and $a$ are as above. Also, for $a<0$ and $1<{ n(p-1) \over 2 } < p,$ an explicit estimate is given for the range of allowed values of $a.$