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

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.

[1]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[2]  L. Herraiz,et al.  Asymptotic behaviour of solutions of some semilinear parabolic problems , 1999 .

[3]  B. Kawohl.  When are solutions to nonlinear elliptic boundary value problems convex , 1985 .

[4]  G. Sansone.  Equazioni differenziali nel campo reale , 1948 .

[5]  J H Lee,et al.  Receptor-mediated endocytosis of soluble and membrane-tethered Sonic hedgehog by Patched-1. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[6]  S. Shvartsman,et al.  Self-similar dynamics of morphogen gradients. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  L. Peletier,et al.  A very singular solution of the heat equation with absorption , 1986 .

[8]  Gianni Dal Maso,et al.  An Introduction to [gamma]-convergence , 1993 .

[9]  Otared Kavian,et al.  Variational problems related to self-similar solutions of the heat equation , 1987 .

[10]  A. Friedman,et al.  Nonlinear Parabolic Equations Involving Measures as Initial Conditions , 1981 .

[11]  Gregory T. Reeves,et al.  Quantitative models of developmental pattern formation. , 2006, Developmental cell.

[12]  A. Kicheva,et al.  Morphogen gradient formation. , 2009, Cold Spring Harbor perspectives in biology.

[13]  G. I. Barenblatt.  Scaling: Self-similarity and intermediate asymptotics , 1996 .

[14]  O. Kavian,et al.  Large time behavior of solutions of a dissipative semilnear heat equation , 1995 .

[15]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[16]  J. Bricmont,et al.  Stable Non-Gaussian Diffusive Profiles , 1993 .

[17]  Peter V. Gordon,et al.  Local kinetics of morphogen gradients , 2011, Proceedings of the National Academy of Sciences.

[18]  L. Peletier,et al.  Singular solutions of the heat equation with absorption , 1985 .

[19]  G. Struhl,et al.  Dual Roles for Patched in Sequestering and Transducing Hedgehog , 1996, Cell.

[20]  C. Eugene Wayne,et al.  Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains , 1997 .

[21]  Naama Barkai,et al.  Self-enhanced ligand degradation underlies robustness of morphogen gradients. , 2003, Developmental cell.

[22]  Qing Nie,et al.  The measure of success: constraints, objectives, and tradeoffs in morphogen-mediated patterning. , 2009, Cold Spring Harbor perspectives in biology.

[23]  L. Véron,et al.  Large time behaviour of the solutions of a semilinear parabolic equation in RN , 1984 .

[24]  Laurent Véron.  A note on maximal solutions of nonlinear parabolic equations with absorption , 2011, Asymptot. Anal..

[25]  J. Bricmont,et al.  Stable nongaussian diffusive profiles , 1996 .

[26]  David M. Umulis,et al.  The Intersection of Theory and Application in Elucidating Pattern Formation in Developmental Biology. , 2009, Mathematical modelling of natural phenomena.

[27]  Cyrill B. Muratov,et al.  Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction‐diffusion equations invading an unstable equilibrium , 2004 .

[28]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[29]  Antoinette Tordesillas,et al.  Complex networks in confined comminution. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  V. Galaktionov,et al.  ON ASYMPTOTIC “EIGENFUNCTIONS” OF THE CAUCHY PROBLEM FOR A NONLINEAR PARABOLIC EQUATION , 1986 .

[31]  Alfonso Martinez Arias,et al.  Molecular Principles of Animal Development , 2002 .