Noncoercive diffusion equations with Radon measures as initial data

We study Radon measure-valued solutions of the Cauchy-Dirichlet problem for $pa_t u = Delta phi (u)$ for a continuous, nondecreasing, at most powerlike $phi$. We prove well-posedness and regularity results, which depend on whether or not the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of $phi$), and on suitable {em compatibility conditions}.