# American Institute of Mathematical Sciences

July  2013, 12(4): 1769-1782. doi: 10.3934/cpaa.2013.12.1769

## Propagation of singularities of nonlinear heat flow in fissured media

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

Received  June 2011 Revised  June 2012 Published  November 2012

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 $\Gamma$.
Citation: Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769
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