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

Abstract

Consider the anisotropic porous medium equation, $u_t=\sum\limits_{i=1}^n(u^{m_i})_{x_ix_i},$ where $m_i>0, (i=1,2,...,n)$ satisfying $\min\limits_{1\le i\le n}\{m_i\}\le 1,$ $\sum\limits_{i=1}^nm_i>n-2,$ and $\max\limits_{1\le i\le n}\{m_i\}\le \frac{1}{n}(2+\sum\limits_{i=1}^nm_i).$ 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.