Solutions of the anisotropic porous medium equation in RnRn under an L1L1-initial value

Consider the anisotropic porous medium equation, ut=∑i=1n(umi)xixi, where mi>0,(i=1,2,…,n) satisfying max1⩽i⩽n{mi}⩽1max1⩽i⩽n{mi}⩽1, ∑i=1nmi>n-2, and max1⩽i⩽n{mi}⩽1/n(2+∑i=1nmi). Assuming that the initial data belong only to L1(Rn)L1(Rn), we establish the existence and uniqueness of the solution for the Cauchy problem in the space, C([0,∞),L1(Rn))∩C(Rn×(0,∞))∩L∞(Rn×[ε,∞))C([0,∞),L1(Rn))∩C(Rn×(0,∞))∩L∞(Rn×[ε,∞)), where ε>0ε>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∞(Rn)L∞(Rn) as time goes to infinity.