Propagation of Singularities of Nonlinear Heat Flow in Fissured Media

TL;DR

This paper studies how singularities propagate in a nonlinear heat equation with degenerate absorption along a curve, revealing how media properties influence heat behavior in fissured environments.

Contribution

It introduces a model for nonlinear heat flow with degenerate absorption along a curve, analyzing singularity propagation in fissured media.

Findings

Singularities propagate along specific space-time curves.

Degenerate absorption significantly affects heat distribution.

The model captures key features of fissured media heat flow.

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-\Delta u+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.