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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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In this paper we investigate the semilinear partial differential equation ut = -fuxxxx - uxx + 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 LX 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
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Bartuccelli, M.
Gourley, S.
Ilyin, A. A.
Date : 8 June 2002
Uncontrolled Keywords : Interpolation, Inequalities, Positivity, Attractors, Fractal Dimension
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.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:41
Last Modified : 16 Jan 2019 16:22

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