Radon measure-valued solutions of quasilinear parabolic equations

Abstract

We discuss some recent results concerning Radon measure-valued solutions of the Cauchy–Dirichlet problem for $\\partial_t u = \\Delta\\phi(u)$. The function $\\phi$ is continuous, nondecreasing, with a growth at most powerlike. Well-posedness and regularity results are described, which depend on whether the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of $\\phi$), and on suitable compatibility conditions at $\\pm\\infty$.