A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition

Details A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition

2007-11-20

arxiv.org/abs/0711.3042v1

Mathematics

Analysis of PDEs

25 pages, submitted

Scientific paper

We study the existence, uniqueness and regularity of solutions of the equation $f_t = \Delta_p f = \text{div} (|Df|^{p-2} Df)$ under over-determined boundary conditions $f = 0$ and $|Df| = 1$. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until it vanishes identically. Furthermore, the free-boundary of the support of $f$ is smooth for all positive time.

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