A sharp estimate and change on the dimension of the attractor for singular semilinear parabolic equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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A sharp estimate and change on the dimension of the attractor for singular semilinear parabolic equations

Authors:

Nikos I Karachalios Nikos B Zographopoulos

Institution:

(1) Department of Mathematics, University of the Aegean, Karlovassi GR 83200 Samos, Greece;(2) Department of Mathematics, Technical University of Crete, Chania, GR, 73100 Crete, Greece

Abstract:

We consider the semilinear reaction diffusion equation

$\partial_{t}\phi - \nu\triangle\phi - V{x}\phi + f{\phi} = 0, \nu > 0$

, in a bounded domain $\Omega \subset {\mathbb{R}}^{N}$ . We assume the standard “Allen-Cahn-type” nonlinearity, while V is either the inverse square potential $V (x) = \delta |x|^{-2}$ or the borderline potential $V (x) = \delta {\rm dist}(x, \partial\Omega)^{-2}, \delta \geq 0$ (thus including the classical Allen-Cahn-type equation as a special case when $\delta = 0$ ). In the subcritical cases $\delta = 0, N \geq 1$ and $0 < \mu := \frac{\delta}{\nu} < \mu^{*} , N \geq 3$ where $\mu^{*}$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new estimate on the dimension of the global attractor. This estimate comes out by an improved lower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math. Soc. 131 (2003), 631–636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio of the volume to the moment of inertia of Ω on which the dimension of the attractor may considerably change. Consideration is also given on the finite dimensionality of the global attractor in the critical case $\mu =\mu^{*}$ Received: 7 May 2008

Keywords:

" target="_blank"> Semilinear parabolic equation Allen-Cahn equation singular potential Hardy inequality attractors Hausdorff dimension fractal dimension

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