On reaction processes with a logarithmic-diffusion

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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<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(α−u)(1−u) we show that for α<1/3 the system may evolve into a TW, an equilibrium, an expanding formation or to collapse. The 1/3<α regime admits either a cooling TW or a collapse. Few other transport processes are outlined in the appendix.

Original languageEnglish
Pages (from-to)94-101
Number of pages8
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number2
StatePublished - 15 Jan 2017


  • Attractors
  • Explicit solutions
  • Fast diffusion
  • Quadratic and bi-stable reactions
  • Solution's extinction


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