Radon measure-valued solutions to a class of noncoercive diffusion equations with singular initial data

Initial-boundary value problems for nonlinear parabolic equations $u_t = \Delta \phi(u)$ with a Radon measure as initial condition have been widely investigated, in general looking for solutions which for positive times take values in some function space. On the other hand, if the diffusivity degenerates too fast at infinity, it is well known that function-valued solutions may not exist, and in these singular cases it looks very natural to consider Radon measure-valued solutions.
The aim of this talk is to address existence and regularity results in the above framework, depending on whether or not the initial data charge sets of suitable capacity (determined by the growth order of $\phi$), and on suitable compatibility conditions, describing the behaviour of the singular part of solutions. The diffusion function $\phi$ is only assumed to be continuous, nondecreasing and at most powerlike: no assumptions about existence or estimates from below of the diffusivity $\phi'$ are made (except for some regularization results). The proof of existence is constructive and, in particuar, relies on a suitable approximation of the initial measure. Finally, the possible occurrence or lack of instantaneous $M-L^1$ regularizing effects for the constructed solutions, as well as partial uniqueness results, will also be discussed.