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Fundamental Solution of the Anisotropic Porous Medium Equation

Authors:	Bin Heng Song Huai Yu Jian

Institution:	(1) School of Software, Tsinghua University, Beijing 100084, P. R. China;(2) Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China

Abstract:	We establish the existence of fundamental solutions for the anisotropic porous medium equation, $u_{t} = {\sum\nolimits_{i = 1}^n {{\left( {u^{{m_{i} }} } \right)}x_{i} x_{i} } }$ in ℜⁿ × (0,∞), where m ₁,m ₂, . . . , and m _n are positive constants satisfying $\min _{{1 \leqslant i \leqslant n}} {\left\{ {m_{i} } \right\}} \leqslant 1,{\sum\nolimits_{i = 1}^n {m_{i} > n - 2,} }$ and $\max _{{1 \leqslant i \leqslant n}} {\left\{ {m_{i} } \right\}} \leqslant \frac{1} {n}{\left( {2 + {\sum\nolimits_{i = 1}^n {m_{i} } }} \right)}.$ The research is supported by National 973–Project and the Trans-Century Training Programme Foundation for the Talents by the Ministry of Education

Keywords:	Yhndamental solution Anisotropic porous medium equation Degenerate parabolic equation
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