LENGTH SCALES IN SOLUTIONS OF A GENERALIZED DIFFUSION-MODEL

In the theory surrounding dissipative partial differential equations, length scales are one of the most important dynamical concepts for pro perly understanding the spatio-temporal patterns of dissipative flows. In this paper we investigate a set of length scales for the solutions of the dissipative partial differential equation u(t) = -alpha del(4) u - beta del(2)u + gamma del(2)u(2q+1) + lambda u - delta u\u\ on peri odic boundary conditions and for one spatial dimension. Our length sca les are based on ratios of norms, which involve a set of differential inequalities proved for the above equation. Lower bounds are derived f or the time averages of these length scales. The model in the case q = 1, was first studied in the context of population dynamics in a paper by D. S. Cohen and J. D. Murray.