Propagation of Singularities of Nonlinear Heat Flow in Fissured Media

Abstract

In this paper we investigate the propagation of singularities in a nonlinear parabolic equation with strong absorption when the absorption potential is strongly degenerate following some curve in the $(x,t)$ space. As a very simplified model, we assume that the heat conduction is constant but the absorption of the media depends stronly of the characteristic of the media. More precisely we suppose that the temperature $u$ is governed by the following equation \label{I-1} \partial_{t}u-Δu+h(x,t)u^p=0\quad \text{in}Q_{T}:=R^N\times (0,T) where $p>1$ and $h\in C({\bar{Q}}_T)$. We suppose that $h(x,t)>0$ except when $(x,t)$ belongs to some space-time curve.

To appear in Comm. Pure Appl. Anal