Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations

We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: (i) the concentrated part of the solution with respect to the Newtonian capacity is constant; (ii) the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem (P) are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given. © 2013 Springer-Verlag Berlin Heidelberg.