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

TL;DR

This paper investigates the existence, uniqueness, and initial trace of positive solutions to a degenerate p-Laplacian heat equation with absorption, considering unbounded initial data and nonlinearities, including specific models like power-log functions.

Contribution

It provides new conditions for existence and uniqueness of solutions with unbounded initial data and explores initial trace characterization for a broad class of nonlinearities.

Findings

Established sufficient conditions for solution existence and uniqueness.

Analyzed the limit behavior of solutions as initial data magnitude grows.

Proved that solutions admit initial traces in the space of positive Borel measures.

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 $\gb\geq 0$.