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 ${\partial }_{t}u=\Delta \varphi (u)$. The function $\varphi$ 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 $\varphi$), and on suitable compatibility conditions at $\pm \infty$.