On the Cauchy Problem for a Degenerate Parabolic Equation

Abstract

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

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.