Interior and boundary continuity of weak solutions of degenerate parabolic equations

Abstract:

In this paper we consider degenerate parabolic equations of the form $({\ast })$ \[ \beta {(u)_t} - \operatorname {div} A(x, t, u, {u_x}) + B(x, t, u, {u_x}) \ni 0\] where $A$ and $B$ are, respectively, vector and scalar valued Baire functions defined on $U \times {R^1} \times {R^n}$, where $U$ is an open subset of ${R^{n + 1}}(x, t)$. The functions $A$ and $B$ are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation $\beta \subset {R^1} \times {R^1}$ so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of $({\ast })$ are continuous throughout $U$. In the event that $U = \Omega \times (0, T)$ where $\Omega$ is an open set of ${R^n}$, it is also shown that a weak solution is continuous at $({x_0},{t_0}) \in \partial \Omega \times (0, T)$ provided ${x_0}$ is a regular point for the Laplacian on $\Omega$.