No abstract

We study nonnegative solutions of the Cauchy problemwhere u 0 is a Radon measure and ϕ ∶ [0, ∞) ↦ R is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on ϕ, we prove their uniqueness if the singular part of u 0 is a finite superposition of Dirac masses.In terms of the behaviour of ϕ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case ϕ(u) = u this happens for all times). In the latter case we describe the evolution of the singular parts.

We discuss some properties of a forward-backward parabolic problem that arises in models of phase transition in which two stable phases are separated by an interface. Here we consider a formulation of the problem that comes from a Sobolev approximation of it. In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data

No abstract

No abstract