Localization of solutions of anisotropic parabolic equations

✍ Scribed by Stanislav Antontsev; Sergey Shmarev

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 357 KB
• Volume: 71
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2008.11.025

✦ Synopsis

We study the localization properties of solutions of the Dirichlet problem for the anisotropic parabolic equations

arise from the mathematical description of diffusion processes. It is shown that if the equation combines the directions of slow diffusion for which p i > 2 and the directions of fast or linear diffusion corresponding to p i ∈ (1, 2) or p = 2, then the solutions may simultaneously display the properties intrinsic for the solutions of isotropic equations of fast or slow diffusion. Under the assumptions that f ≡ 0 for t ≥ t f and u 0 ≡ 0, f ≡ 0 for x 1 > s we show, on the one hand, that the solution vanishes in a finite time

2 and, on the other hand, that the support of the same solution never reaches the plane x 1 = s + , provided that 1 n-1 ≥ 1 n-1 n i=2