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

  • 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.

    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

    Related Papers:

    [1] 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. Networks and Heterogeneous Media, 2012, 7(4): 767-780. doi: 10.3934/nhm.2012.7.767
    [2] Bendong Lou . Self-similar solutions in a sector for a quasilinear parabolic equation. Networks and Heterogeneous Media, 2012, 7(4): 857-879. doi: 10.3934/nhm.2012.7.857
    [3] Junlong Chen, Yanbin Tang . Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure. Networks and Heterogeneous Media, 2023, 18(3): 1118-1177. doi: 10.3934/nhm.2023049
    [4] Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch . Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16(2): 283-315. doi: 10.3934/nhm.2021007
    [5] Panpan Xu, Yongbin Ge, Lin Zhang . High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source. Networks and Heterogeneous Media, 2023, 18(4): 1471-1492. doi: 10.3934/nhm.2023065
    [6] Kota Kumazaki, Toyohiko Aiki, Adrian Muntean . Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect. Networks and Heterogeneous Media, 2023, 18(1): 80-108. doi: 10.3934/nhm.2023004
    [7] Benjamin Contri . Fisher-KPP equations and applications to a model in medical sciences. Networks and Heterogeneous Media, 2018, 13(1): 119-153. doi: 10.3934/nhm.2018006
    [8] Thomas Geert de Jong, Georg Prokert, Alef Edou Sterk . Reaction–diffusion transport into core-shell geometry: Well-posedness and stability of stationary solutions. Networks and Heterogeneous Media, 2025, 20(1): 1-14. doi: 10.3934/nhm.2025001
    [9] Linglong Du . Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks and Heterogeneous Media, 2018, 13(4): 549-565. doi: 10.3934/nhm.2018025
    [10] Hirotada Honda . Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12(1): 25-57. doi: 10.3934/nhm.2017002
  • 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] G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics," Cambridge University Press, 1996.
    [2] H. Brézis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97.
    [3] H. Brezis, L. A. Peletier and D. Terman, A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal., 95 (1986), 185-209. doi: 10.1007/BF00251357
    [4] J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles, Nonlinear Anal., 26 (1996), 583-593. doi: 10.1016/0362-546X(94)00300-7
    [5] Y. Chen and G. Struhl, Dual roles for patched in sequestering and transducing hedgehog, Cell, 87 (1996), 553-563.
    [6] G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8
    [7] A. Eldar, D. Rosin, B. Z. Shilo and N. Barkai, Self-enhanced ligand degradation underlies robustness of morphogen gradients, Devel. Cell, 5 (2003), 635-646.
    [8] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0
    [9] M. Escobedo and O. Kavian, Asymptotic behaviour of positive solutions of a nonlinear heat equation, Houston J. Math., 14 (1988), 39-50.
    [10] M. Escobedo, O. Kavian and H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations, 20 (1995), 1427-1452. doi: 10.1080/03605309508821138
    [11] V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskiĭ, Asymptotic "eigenfunctions'' of the Cauchy problem for a nonlinear parabolic equation, Mat. Sb. (N.S.), 126 (1985), 435-472.
    [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.
    [13] A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in $R^N$, J. Differential Equations, 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1
    [14] P. V. Gordon, C. Sample, A. M. Berezhkovskii, C. B. Muratov and S. Y. Shvartsman, Local kinetics of morphogen gradients, Proc. Natl. Acad. Sci. US., 108 (2011), 6157-6162.
    [15] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49-105. doi: 10.1016/S0294-1449(99)80008-0
    [16] J. P. Incardona, J. H. Lee, C. P. Robertson, K. Enga, R. P. Kapur and H. Roelink, Receptor-mediated endocytosis of soluble and membrane-tethered sonic hedgehog by patched-1, Proc. Natl. Acad. Sci. USA, 97 (2000), 12044-12049.
    [17] S. Kamin and L. A. Peletier, Singular solutions of the heat equation with absorption, Proc. Amer. Math. Soc., 95 (1985), 205-210. doi: 10.2307/2044513
    [18] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations, 10 (1985), 1213-1225. doi: 10.1080/03605308508820404
    [19] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, New York, 1980.
    [20] A. D. Lander, W. C. Lo, Q. Nie and F. Y. Wan, The measure of success: constraints, objectives, and tradeoffs in morphogen-mediated patterning, Cold Spring Harbor Perspectives in Biology, 1 (2009), a002022.
    [21] M. Lucia, C. B. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Commun. Pure Appl. Math., 57 (2004), 616-636. doi: 10.1002/cpa.20014
    [22] A. Martinez-Arias and A. Stewart, "Molecular Principles of Animal Development," Oxford University Press, New York, 2002.
    [23] C. B. Muratov, P. V. Gordon and S. Y. Shvartsman, Self-similar dynamics of morphogen gradients, Phys. Rev. E, 84 (2011), 1-4. 041916.
    [24] L. Oswald, Isolated positive singularities for a nonlinear heat equation, Houston J. Math., 14 (1988), 543-572.
    [25] H. G. Othmer, K. Painter, D. Umulis and C. Xue, The intersection of theory and application in elucidating pattern formation in developmental biology, Math. Model. Nat. Phenom., 4 (2009), 3-82. doi: 10.1051/mmnp/20094401
    [26] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5
    [27] G. T. Reeves, C. B. Muratov, T. Schüpbach and S. Y. Shvartsman, Quantitative models of developmental pattern formation, Devel. Cell, 11 (2006), 289-300.
    [28] G. Sansone, "Equazioni Differenziali nel Campo Reale," 2. Nicola Zanichelli, Bologna, 1949. 2d ed.
    [29] L. Veron, A note on maximal solutions of nonlinear parabolic equations with absorption, arXiv:0906.0669v2 [math.AP], 2011.
    [30] O. Wartlick, A. Kicheva and M. Gonzalez-Gaitan, Morphogen gradient formation, Cold Spring Harbor Perspectives in Biology, 1 (2009), a001255.
    [31] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal., 138 (1997), 279-306. doi: 10.1007/s002050050042
  • This article has been cited by:

    1. Xiaowei Liu, Jin Zhang, Asymptotic behavior of solutions of a reaction–diffusion equation with inhomogeneous Robin boundary condition and free boundary condition, 2016, 28, 14681218, 126, 10.1016/j.nonrwa.2015.07.019
    2. Peter V. Gordon, Cyrill B. Muratov, Eventual Self-similarity of Solutions for the Diffusion Equation with Nonlinear Absorption and a Point Source, 2015, 47, 0036-1410, 2903, 10.1137/140974997
    3. Peter V. Gordon, Cyrill B. Muratov, Stanislav Y. Shvartsman, Local accumulation times for source, diffusion, and degradation models in two and three dimensions, 2013, 138, 0021-9606, 104121, 10.1063/1.4793985
    4. Cyrill B. Muratov, Xiaodong Yan, Uniqueness of one-dimensional Néel wall profiles, 2016, 472, 1364-5021, 20150762, 10.1098/rspa.2015.0762
    5. Milena Chermisi, Cyrill B Muratov, One-dimensional Néel walls under applied external fields, 2013, 26, 0951-7715, 2935, 10.1088/0951-7715/26/11/2935
    6. Hamid Teimouri, Anatoly B Kolomeisky, Mechanisms of the formation of biological signaling profiles, 2016, 49, 1751-8113, 483001, 10.1088/1751-8113/49/48/483001
    7. Hamid Teimouri, Anatoly B. Kolomeisky, 2018, Chapter 12, 978-1-4939-8771-9, 199, 10.1007/978-1-4939-8772-6_12
  • Reader Comments
  • © 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4484) PDF downloads(86) Cited by(7)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog