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MiS Preprint

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

Ugur G. Abdulla


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.

MSC Codes:
35K65, 35K55
dirichlet problem, non-smooth domains, nonlinear diffusion, degenerate and singular parabolic equations, boundary regularity

Related publications

2001 Repository Open Access
Ugur G. Abdulla

On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains

In: Journal of mathematical analysis and applications, 260 (2001) 2, pp. 384-403