Propagation of singularities of nonlinear heat flow in fissured media

Andrey Shishkov; Laurent Véron

doi:10.3934/cpaa.2013.12.1769

Article Contents

2013, Volume 12, Issue 4: 1769-1782. Doi: 10.3934/cpaa.2013.12.1769

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Propagation of singularities of nonlinear heat flow in fissured media

Andrey Shishkov^1, and
Laurent Véron^2,

1.
Institute of Applied Mathematics and Mechanics, 83114 Donetsk
2.
Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Université François-Rabelais, 37200 Tours

Received: May 31, 2011

Revised: May 31, 2012

Published: June 2013

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Abstract

Let $\Gamma=\{\gamma(\tau)\in R^N\times [0,T], \gamma(0)=(0,0)\}$ be $C^{0,1}$ -- space-time curve and continuos function $h(x,t)>0$ in $ R^N\times [0,T]\setminus \Gamma (h(x,t)=0$ on $\Gamma$). We investigate the behaviour as $k\to \infty$ of the fundamental solutions $u_k$ of equation $u_t-\Delta u+h(x,t)u^p=0$, $p>1$, satisfying singular initial condition $u_k(x,0)=k\delta_0$. The main problem is whether the limit $u_\infty$ is still a solution of the above equation with isolated point singularity at $(0,0)$, or singularity set of $u_\infty$ contains some part or all $\Gamma$.

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Mathematics Subject Classification: Primary: 35K60; Secondary: 35K55.

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