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; Peter V. Gordon; Cyrill B. Muratov

doi:10.3934/nhm.2012.7.767

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 767-780. doi: 10.3934/nhm.2012.7.767

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

Peter V. Gordon ^{1
,},
Cyrill B. Muratov ^{2
,}

1.
Department of Mathematics, The University of Akron, Akron, OH 44325
2.
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102

Received: 01 January 2012 Revised: 01 August 2012
Primary: 35C06, 35K61, 35B40; Secondary: 35Q92.

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.
- Self-similarity,
- nonlinear diffusion equation,
- morphogen gradients
Citation: Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source[J]. Networks and Heterogeneous Media, 2012, 7(4): 767-780. doi: 10.3934/nhm.2012.7.767

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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.

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