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

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.