On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

We study the Dirichlet problem for the parabolic equation ut = um m > 0, in a bounded, non-cylindrical and non-smooth domain ⊂ N+1 N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) H¨older condition on it plays a crucial role for the boundary continuity of the constructed solution. The H¨older exponent 1 2 is critical as in the classical theory of the one-dimensional heat equation ut = uxx.