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.
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.
|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|
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