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 $t\to \infty$. Furthermore, for a certain class of functions $f$ explicit decay rates are established.