Large time behavior of solutions for parabolic equations with nonlinear gradient terms

Abstract

In this paper we prove the global existence of mild solutions for the semilinear parabolic equation $u_t=\Delta u+a|\nabla u|^q+b|u|^{p-1}u, t>0, x \in \mathbb R^n, n \geq 1,$ $a \in \mathbb R, b \in \mathbb R, p>1+(2/n),$ $(n+2)/(n+1)<q<2$ and $q\leq p(n+2)/(n+p),$ with small initial data with respect to a norm related to the equation. We also prove that some of these global solutions are asymptotic to self--similar solutions of the equations $u_t=\Delta u+\nu a|\nabla u|^q+\mu b|u|^{p-1}u,$ with $\nu, \mu=0$ or $1.$ The values of $\nu$ and $\mu$ depend on the decaying of the initial data and on the position of $q$ with respect to $2p/(p+1).$ Our results apply for the viscous Hamilton--Jacobi equation: $u_t=\Delta u+a|\nabla u|^q$ and hold without sign restriction neither on $a$ nor on the initial data. We prove that if $(n+2)/(n+1)<q<2$ and the initial data behaves, near infinity, like $c|x|^{-\alpha}, (2-q)/(q-1) \leq \alpha<n, c$ is a small constant, then the resulting solution is global. Moreover, if $(2-q)/(q-1)<\alpha<n,$ the solution is asymptotic to a self--similar solution of the linear heat equation. Whereas, if $(2-q)/(q-1)=\alpha<n,$ the solution is asymptotic to a self--similar solution of the viscous Hamilton--Jacobi equation. The asymptotics are given, in particular, in $W^{1,\infty}(\mathbb R^n).$