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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 3, 2001, pp. 677–690
DOI: 10.4171/ZAA/1038

Published online: 2001-09-30

On the Cauchy Problem for a Degenerate Parabolic Equation

Michael Winkler[1]

(1) Universität Paderborn, Germany

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.

Keywords: Degenerate diffusion, large-time behaviour

Winkler Michael: On the Cauchy Problem for a Degenerate Parabolic Equation. Z. Anal. Anwend. 20 (2001), 677-690. doi: 10.4171/ZAA/1038