# On the Cauchy Problem for a Degenerate Parabolic Equation

### Michael Winkler

Universität Paderborn, Germany

## Abstract

Existence and uniqueness of global positive solutions to the degenerate parabolic problem

$u_t = f(u)\Delta u \ \ \mathrm {in} \ \ \mathbb R^n \times (0, \infty)$

$u|_{t=0} = u–0$

with $f \in C^0 ((0, \infty)) \cap C^1 ((0, \infty))$ satisfying $f(0) = 0$ and $f(s) > 0$ for $s > 0$ are investigated. It is proved that, without any further conditions on $f$, decay of $u_0$ in space implies uniform zero convergence of $u(t)$ as $t \rightarrow \infty$. Furthermore, for a certain class of functions $f$ explicit decay rates are established.

## Cite this article

Michael Winkler, On the Cauchy Problem for a Degenerate Parabolic Equation. Z. Anal. Anwend. 20 (2001), no. 3, pp. 677–690

DOI 10.4171/ZAA/1038