## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 36, Number 2 (2007), 311-344.

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

Seifeddine SNOUSSI and Slim TAYACHI

#### 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 $W^{1,\infty}(\mathbb R^n).$

#### Article information

**Source**

Hokkaido Math. J., Volume 36, Number 2 (2007), 311-344.

**Dates**

First available in Project Euclid: 25 June 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1277472806

**Digital Object Identifier**

doi:10.14492/hokmj/1277472806

**Mathematical Reviews number (MathSciNet)**

MR2347428

**Zentralblatt MATH identifier**

1180.35113

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35K65: Degenerate parabolic equations 35K15: Initial value problems for second-order parabolic equations 35B40: Asymptotic behavior of solutions

**Keywords**

Semilinear parabolic equations Global solutions Large time behavior Self-similar solutions Nonlinear gradient term Viscous Hamilton–Jacobi equation

#### Citation

SNOUSSI, Seifeddine; TAYACHI, Slim. Large time behavior of solutions for parabolic equations with nonlinear gradient terms. Hokkaido Math. J. 36 (2007), no. 2, 311--344. doi:10.14492/hokmj/1277472806. https://projecteuclid.org/euclid.hokmj/1277472806