Solution of the anisotropic porous medium equation in $R^n$ under an $L^1$-initial value

Abstract

Consider the anisotropic porous medium equation, $u_t=\sum _{i=1}^n(u^{m_i}{)}_{x_i{x}_i},$ where $m_i>0,(i=1,2,...,n)$ satisfying $\underset{1\le i\le n}{\min }\{m_i\}\le 1,$ $\sum _{i=1}^n{m}_i>n-2,$ and $\underset{1\le i\le n}{\max }\{m_i\}\le \frac{1}{n}(2+\sum _{i=1}^n{m}_i).$ Assuming that the initial data belong only to $L^1({\Re }^n)$, we establish the existence and uniqueness of the solution for the Cauchy problem in the space, $C([0,\infty ),L^1({\Re }^n))\cap C({\Re }^n\times (0,\infty ))\cap L^{\infty }({\Re }^n\times [\epsilon ,\infty )),$ where $\epsilon >0$ may be arbitrary. We also show a comparison principle for such solutions. Furthermore, we prove that the solution converges to zero in the space $L^{\infty }({\Re }^n)$ as the time goes to infinity.

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