## Advances in Differential Equations

- Adv. Differential Equations
- Volume 14, Number 7/8 (2009), 663-684.

### Pointwise decay for the solutions of degenerate and singular parabolic equations

Petri Juutinen and Peter Lindqvist

#### Abstract

We study the asymptotic behavior, as $t\to\infty$, of the solutions to the evolutionary $p$-Laplace equation \[ v_t={\operatorname{div}}( | {\nabla v} | ^{p-2}\nabla v), \] with time-independent lateral boundary values. We obtain the sharp decay rate of $\max_{x\in{\Omega}} | {v(x,t)-u(x)}| $, where $u$ is the stationary solution, both in the degenerate case $p > 2$ and in the singular case $1 < p > 2$. A key tool in the proofs is the Moser iteration, which is applied to the difference $v(x,t)-u(x)$. In the singular case, we construct an example proving that the celebrated phenomenon of finite extinction time, valid for $v(x,t)$ when $u\equiv 0$, does not have a counterpart for $v(x,t)-u(x)$.

#### Article information

**Source**

Adv. Differential Equations, Volume 14, Number 7/8 (2009), 663-684.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867230

**Mathematical Reviews number (MathSciNet)**

MR2527689

**Zentralblatt MATH identifier**

1182.35036

**Subjects**

Primary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 35B40: Asymptotic behavior of solutions

#### Citation

Juutinen, Petri; Lindqvist, Peter. Pointwise decay for the solutions of degenerate and singular parabolic equations. Adv. Differential Equations 14 (2009), no. 7/8, 663--684. https://projecteuclid.org/euclid.ade/1355867230