Positivity and the attractor dimension in a fourth-order reaction-diffusion equation

Bartuccelli, M. V., Gourley, S. A. and Ilyin, A. A. (2002) Positivity and the attractor dimension in a fourth-order reaction-diffusion equation 458. pp. 1431-1446.

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Abstract

In this paper we investigate the semilinear partial differential equation u_t = -fu_xxxx - u_xx + u(1 - u²) with a view, particularly, to obtaining some insight into how one might establish positivity preservation results for equations containing fourth-order spatial derivatives. The maximum principle cannot be applied to such equations. However, progress can be made by employing some very recent 'best possible' interpolation inequalities, due to the third-named author, in which the interpolation constants are both explicitly known and sharp. These are used to estimate the L^X distance between u and 1 during the evolution. A positivity preservation result can be obtained under certain restrictions on the initial datum. We also establish an explicit two-sided estimate for the fractal dimension of the attractor, which is sharp in terms of the physical parameters.

Item Type:	Article
Additional Information:	Published in <Oroccedings of the Royal Society of London>A</i>, <i>458</i>, 1431-1446. © 2002 The Royal Society. All rights reserved.
Uncontrolled Keywords:	Interpolation, Inequalities, Positivity, Attractors, Fractal Dimension
Divisions:	Faculty of Engineering and Physical Sciences > Mathematics
Depositing User:	Mr Adam Field
Date Deposited:	27 May 2010 14:41
Last Modified:	28 Sep 2012 09:50
URI:	http://surrey.eprints.org/id/eprint/1456

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