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MiS Preprint

Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

Ugur G. Abdulla


We investigate the Dirichlet problem for the parablic equation $$ u_t = \Delta u^m, m > 0, $$ in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \ge 2$. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains.

MSC Codes:
35K65, 35K55
dirichlet problem, non-smooth domains, nonlinear diffusion, degenerate and singular parabolic equations, uniqueness and comparison results, l1-contraction, boundary gradient estimates

Related publications

2005 Repository Open Access
Ugur G. Abdulla

Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains

In: Transactions of the American Mathematical Society, 357 (2005) 1, pp. 247-265