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].
|||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.|
|||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.|
|||U.G.Abdulla, Reaction-Diffusion in irregular domains, Journal of Differential Equations, 164(2000), 321-354.|
|||U.G.Abdulla, Reaction-Diffusion in a closed domain formed by irregular curves, Journal of Mathematical Analysis and Applications, 246(2000), 480-492.|
|||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.|