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

Document Type

Article

Publication Date

12-1-2012

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. © American Institute of Mathematical Sciences.

Identifier

84875157138 (Scopus)

Publication Title

Networks and Heterogeneous Media

External Full Text Location

https://doi.org/10.3934/nhm.2012.7.767

e-ISSN

1556181X

ISSN

15561801

First Page

767

Last Page

780

Issue

4

Volume

7

Grant

0908279

Fund Ref

National Science Foundation

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