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

In this paper we investigate the semilinear partial differential equationut = –αuxxxx – 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 L∞ 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.