Initial trace of positive solutions of some nonlinear diffusion equations.

It is well known that any positive solution u of the heat equation in $(0;T)xR^N$ admits an initial trace at $t = 0$ which is a nonnegative Radon measure in $R^N$ which satisfies some exponential growth condition. The situation may completely change if u satisfies a semilinear heat equation with absorption. If the absorption is strong enough, the initial trace exists in the class of general Borel measures. The reverse problem of assigning to a Borel measure a solution of the semilinear heat equation gives rise to huge difficulties. These problems were studied by M. Marcus, L. Veron, J.F. Le Gall, E. Dynkin, S. Kuznetsov, P.T Nguyen, K. Gkikas.