## Abstract

We study formation of patterns in reaction processes with a logarithmic-diffusion: u_{t}=(lnu)_{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. 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(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 language | English |
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Pages (from-to) | 94-101 |

Number of pages | 8 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 381 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jan 2017 |

## Keywords

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