We study the recently proposed convection-diffusion model equation ut + f(u)x = Q(ux)x, with a bounded function Q(ux). We consider both strictly monotone dissipation fluxes with Q1(ux) > 0, and nonmonotone ones such that Q(ux) = ±vux/(1 + u2x), v > 0. The novel feature of these equations is that large amplitude solutions develop spontaneous discontinuities, while small solutions remain smooth at all times. Indeed, small amplitude kink solutions are smooth, while large amplitude kinks have discontinuities (subshocks). It is demonstrated numerically that both continuous and discontinuous travelling waves are strong attractors of a wide classes of initial data. We prove that solutions with a sufficiently large initial data blow up in finite time. It is also shown that if Q(ux) is monotone and unbounded, then ux is uniformly bounded for all times. In addition, we present more accurate numerical experiments than previously presented, which demonstrate that solutions to a Cauchy problem with periodic initial data may also break down in a finite time if the initial amplitude is sufficiently large.