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

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- by Ugur G. Abdulla PDF
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**357**(2005), 247-265 Request permission

## Abstract:

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.

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## Additional Information

**Ugur G. Abdulla**- Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901-6975
- Email: abdulla@fit.edu
- Received by editor(s): July 31, 2000
- Received by editor(s) in revised form: July 21, 2003
- Published electronically: February 27, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**357**(2005), 247-265 - MSC (2000): Primary 35K65, 35K55
- DOI: https://doi.org/10.1090/S0002-9947-04-03464-6
- MathSciNet review: 2098094