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MiS Preprint
39/2000
On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains
Ugur G. Abdulla
Abstract
We study the Dirichlet problem for the parabolic equation $u_i = \Delta u^m, \ m>0$ in a bounded, non-cylindrical and non-smooth domain $\Omega \subset \mathbb{R}^{N=1}$, $N\geq 2$. 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 $\frac{1}{2}$ is critical as in the classical theory of the one-dimensional heat equation $u_i =u_{xx'}$. This research was supported by the Alexander von Humboldt Foundation.
