On reaction processes with a logarithmic-diffusion

We study formation of patterns in reaction processes with a logarithmic-diffusion: u(t) = (In u)(xx) + R(u). For the generic R = u(1- u) case the problem of travelling waves, TW, is mapped into a linear one with the propagation speed lambda selected by a boundary condition, b.c. at the far away upstream. Dirichlet b.c. relaxes the process into a steady state, whereas convective b.c.u(x) + hu = 0, leads the system into a heating (cooling) TW for h < 1 (1 < h) or, if h =1, into an equilibrium. We derive explicit solutions of symmetrically expanding waves and of formations which collapse in a finite time. Both are shown to be attractors of classes of initial excitations. For a bi-stable reaction R =-u(alpha- u)(1- u) we show that for a < 1/3 the system may evolve into a TW, an equilibrium, an expanding formation or to collapse. The 1/3 < alpha regime admits either a cooling TW or a collapse. Few other transport processes are outlined in the appendix. (C) 2016 Elsevier B.V. All rights reserved.