Initial trace of positive solutions of a class of degenerate heat equation with absorption

Abstract

We study the initial value problem with unbounded nonnegative functions or measures for the equation $ \prt_tu-\Gd_p u+f(u)=0$ in $\BBR^N\ti(0,\infty)$ where $p>1$, $\Gd_p u = \text{div}(\abs {\nabla u}^{p-2} \nabla u)$ and $f$ is a continuous, nondecreasing nonnegative function such that $f(0)=0$. In the case $p>\frac{2N}{N+1}$, we provide a sufficient condition on $f$ for existence and uniqueness of the solutions satisfying the initial data $k\gd_0$ and we study their limit when $k\to \infty$ according $f^{-1}$ and $F^{-1/p}$ are integrable or not at infinity, where $F(s)=\int_0^s f(\gs)d\gs$. We also give new results dealing with non uniqueness for the initial value problem with unbounded initial data. If $p>2$, we prove that, for a large class of nonlinearities $f$, any positive solution admits an initial trace in the class of positive Borel measures. As a model case we consider the case $f(u)=u^\ga \ln^\gb(u+1)$, where $\ga>0$ and $\gb\geq 0$.