Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains
Ugur G. Abdulla · Transactions of the American Mathematical Society · 2004
We investigate the Dirichlet problem for the parablic equation \[ u_t = \Delta u^m, m > 0, \] in a non-smooth domain $\Omega \subset \mathbb {R}^{N+1}, N \geq 2$. 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.