. A P ] 2 9 N ov 2 00 6 Solutions of the Anisotropic Porous Medium Equation in R n under L 1 - initial Value ∗

Consider the anisotropic porous medium equation, ut = n ∑ i=1 (um)xixi , where mi > 0, (i = 1, 2, · · · , n) satisfying min 1≤i≤n {mi} ≤ 1, n ∑ i=1 mi > n−2, and max 1≤i≤n {mi} ≤ 1 n (2+ n ∑ i=1 mi). Assuming that the initial data belong only to L(R), we establish the existence and uniqueness of the solution for the Cauchy problem in the space, C([0,∞), L(R))∩C(R× (0,∞)) ∩ L(R × [ε,∞)), where ε > 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(R) as the time goes to infinity.