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

AbstractWe study the Dirichlet problem for the parabolic equation ut=Δum,m>0, in a bounded, non-cylindrical and non-smooth domain Ω⊂RN+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) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 12 is critical as in the classical theory of the one-dimensional heat equation ut=uxx