Breakdown in Burgess-type equations with saturating dissipation fluxes

We study the recently proposed convection-diffusion model equation u(t), f(u) Q(u(x))(x), with a bounded function Q(u(x)). We consider both strictly monotone dissipation fluxes with Q(1)(u(x)) > 0. and nonmonotone ones such that Q(u(x)) = +/-vu(x)/(l + u(x)(2)), v > 0. The novel feature of these e quations is that large amplitude solutions develop spontaneous discontinuit ies, while small solutions remain smooth at all times. Indeed, small amplit ude kink solutions are smooth, while large amplitude kinks have discontinui ties (subshocks). It is demonstrated numerically that both continuous and d iscontinuous travelling waves are strong attractors of a wide classes of in itial data. We prove that solutions with a sufficiently large initial data blow up in finite time. It is also shown that if Q(u(x)) is monotone and un bounded, then u, is uniformly bounded for all times. In addition, we presen t more accurate numerical experiments than previously presented, which demo nstrate that solutions to a Cauchy problem with periodic initial data may a lso break down in a finite time if the initial amplitude is sufficiently la rge. AMS classification scheme numbers: 35Kxx, 35Qxx, 65Mxx.