Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

Peter V. Gordon, Cyrill B. Muratov

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.

Original languageEnglish (US)
Pages (from-to)767-780
Number of pages14
JournalNetworks and Heterogeneous Media
Volume7
Issue number4
DOIs
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Engineering(all)
  • Computer Science Applications
  • Applied Mathematics

Keywords

  • Morphogen gradients
  • Nonlinear diffusion equation
  • Self-similarity

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