The p-Laplace heat equation with a source term : self-similar solutions revisited
Abstract
We study the self-similar solutions of any sign of the equation ut-div(|∇u|p-2∇u)=|u|q-1u, in RN, where p,q>1. We extend the results of Haraux-Weissler obtained for p=2 to the case q>p-1>0. In particular we study the existence of slow or fast decaying solutions. For given t>0, the fast solutions u(t,.) have a compact support in RN when p>2, and |x|p/(2-p)u(t,x) is bounded at infinity when p<2. We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p-1+p/N and the critical Sobolev exponent q∗, we study the existence of positive solutions, or the number of the zeros of u(t,.). We prove that any solution u(t,.) is oscillatory when p<2 and q is closed to 1.