The p-Laplace heat equation with a source term : self-similar solutions revisited

TL;DR

This paper investigates self-similar solutions of a p-Laplace heat equation with a source term, extending previous results to a broader range of parameters, and analyzes their existence, decay, support, and oscillatory behavior.

Contribution

It generalizes earlier findings for p=2 to cases where q>p-1>0, detailing solution behaviors, existence criteria, and oscillation properties for the nonlinear p-Laplace heat equation.

Findings

Existence of slow and fast decaying solutions.

Compact support for fast solutions when p>2.

Oscillatory solutions when p<2 and q near 1.

Abstract

We study the self-similar solutions of any sign of the equation u_{t}-div(|∇u|^{p-2}∇u)=|u|^{q-1}u, in R^{N}, 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 R^{N} 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.