-contraction estimation in general non-smooth domains. Applications
to the problem about the evolution of interfaces for the porous medium
equation will be discussed. Similar one-dimensional results are published
recently in [3-5].
Ugur G. Abdulla, Max-Planck Institute for Mathematics in the Sciences, Leipzig
The new results of the author ([1,2]) on the theory of nonlinear parabolic equations in non-smooth
domains will be presented. The model example is the Dirichlet problem for the
nonlinear diffusion equation in a non-cylindrical domain with non-smooth and
characteristic lateral boundary manifold. We introduce the 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) Holder condition on it plays
a crucial role for the boundary continuity of the constructed solution. The
Holder exponent 1/2 is critical as in the classical theory of the
one-dimensional heat equation. Under the similar minimal conditions on the
boundary we prove also uniqueness and comparison results. In particular, we
prove -contraction estimation in general non-smooth domains. Applications
to the problem about the evolution of interfaces for the porous medium
equation will be discussed. Similar one-dimensional results are published
recently in [3-5].
| [1] | U.G.Abdulla, On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, Preprint 39, 2000, MPI for Mathematics in the Sciences, Leipzig. To appear in the Journal of Mathematical Analysis and Applications in 2001. |
| [2] | U.G.Abdulla, Well-posedness of the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, Preprint 40, 2000, MPI for Mathematics in the Sciences, Leipzig. Submitted to Transaction of AMS. |
| [3] | U.G.Abdulla, Reaction-Diffusion in irregular domains, Journal of Differential Equations, 164(2000), 321-354. |
| [4] | U.G.Abdulla, Reaction-Diffusion in a closed domain formed by irregular curves, Journal of Mathematical Analysis and Applications, 246(2000), 480-492. |
| [5] | U.G.Abdulla and J.R.King, Interface development and local solutions to reaction-diffusion equations, SIAM J. Math. Anal., 32, 2 (2000), 235-260. |
| E-Mail: | Ugur.Abdulla@mis.mpg.de |