JournalszaaVol. 20, No. 3pp. 677–690

On the Cauchy Problem for a Degenerate Parabolic Equation

  • Michael Winkler

    Universität Paderborn, Germany
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Abstract

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

ut=f(u)Δu  in  Rn×(0,)u_t = f(u)\Delta u \ \ \mathrm {in} \ \ \mathbb R^n \times (0, \infty)
ut=0=u0u|_{t=0} = u–0

with fC0((0,))C1((0,))f \in C^0 ((0, \infty)) \cap C^1 ((0, \infty)) satisfying f(0)=0f(0) = 0 and f(s)>0f(s) > 0 for s>0s > 0 are investigated. It is proved that, without any further conditions on ff, decay of u0u_0 in space implies uniform zero convergence of u(t)u(t) as tt \rightarrow \infty. Furthermore, for a certain class of functions ff 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