MiS Preprint Repository

We study the Dirichlet problem for the parabolic equation
$u_i=\mathrm{\Delta }{u}^m,m>0$
in a bounded, non-cylindrical and non-smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^{N=1}$, $N\ge 2$. 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 $u_i=u_{x{x}^{\prime }}$.
This research was supported by the Alexander von Humboldt Foundation.