SOLUTIONS OF THE POROUS MEDIUM EQUATION IN RN UNDER OPTIMAL CONDITIONS ON INITIAL VALUES

It is shown that the initial-value problem ut = A(Iul-I u), u(x,0) = u0 (x), where m> 1, has a solution on RN x [0,T) for some T> 0 if-(-_2+N) R 1 f(IxR} U0 (x)Idx is bounded independently of R) 1. The restriction on u0 is known to be necessary as well by recent results of Aronson and Caffarelli, so this theorem is the best possible. Many supplementary results give refined estimates on the solution under various conditions on u0, establish uniqueness within the existence class, allow U0 to be a Radon measure, establish continuous dependence on u0 in various spaces, etc