On reaction processes with a logarithmic-diffusion

We study formation of patterns in reaction processes with a logarithmic-diffusion: ut=(ln⁡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 λ 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. ux+hu=0, leads the system into a heating (cooling) TW for h<1 (1