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On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

✍ Scribed by Ugur G Abdulla

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 154 KB
Volume: 260
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.2001.7458

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✦ Synopsis

We study the Dirichlet problem for the parabolic equation u t = u m m > 0, in a bounded, non-cylindrical and non-smooth domain ⊂ N+1 N ≥ 2. Existence and boundary regularity results are established. We introduce a 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) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 1 2 is critical as in the classical theory of the one-dimensional heat equation u t = u xx .

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