# American Institute of Mathematical Sciences

December  2012, 7(4): 767-780. doi: 10.3934/nhm.2012.7.767

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

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

Received  January 2012 Revised  August 2012 Published  December 2012

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. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
##### References:
 [1] G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics,", Cambridge University Press, (1996).   Google Scholar [2] H. Brézis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions,, J. Math. Pures Appl., 62 (1983), 73.   Google Scholar [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.  doi: 10.1007/BF00251357.  Google Scholar [4] J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles,, Nonlinear Anal., 26 (1996), 583.  doi: 10.1016/0362-546X(94)00300-7.  Google Scholar [5] Y. Chen and G. Struhl, Dual roles for patched in sequestering and transducing hedgehog,, Cell, 87 (1996), 553.   Google Scholar [6] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar [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.   Google Scholar [8] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal., 11 (1987), 1103.  doi: 10.1016/0362-546X(87)90001-0.  Google Scholar [9] M. Escobedo and O. Kavian, Asymptotic behaviour of positive solutions of a nonlinear heat equation,, Houston J. Math., 14 (1988), 39.   Google Scholar [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.  doi: 10.1080/03605309508821138.  Google Scholar [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.   Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).   Google Scholar [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.  doi: 10.1016/0022-0396(84)90042-1.  Google Scholar [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.   Google Scholar [15] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49.  doi: 10.1016/S0294-1449(99)80008-0.  Google Scholar [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.   Google Scholar [17] S. Kamin and L. A. Peletier, Singular solutions of the heat equation with absorption,, Proc. Amer. Math. Soc., 95 (1985), 205.  doi: 10.2307/2044513.  Google Scholar [18] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?,, Comm. Partial Differential Equations, 10 (1985), 1213.  doi: 10.1080/03605308508820404.  Google Scholar [19] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).   Google Scholar [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).   Google Scholar [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.  doi: 10.1002/cpa.20014.  Google Scholar [22] A. Martinez-Arias and A. Stewart, "Molecular Principles of Animal Development,", Oxford University Press, (2002).   Google Scholar [23] C. B. Muratov, P. V. Gordon and S. Y. Shvartsman, Self-similar dynamics of morphogen gradients,, Phys. Rev. E, 84 (2011), 1.   Google Scholar [24] L. Oswald, Isolated positive singularities for a nonlinear heat equation,, Houston J. Math., 14 (1988), 543.   Google Scholar [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.  doi: 10.1051/mmnp/20094401.  Google Scholar [26] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar [27] G. T. Reeves, C. B. Muratov, T. Schüpbach and S. Y. Shvartsman, Quantitative models of developmental pattern formation,, Devel. Cell, 11 (2006), 289.   Google Scholar [28] G. Sansone, "Equazioni Differenziali nel Campo Reale,", 2. Nicola Zanichelli, 2 (1949).   Google Scholar [29] L. Veron, A note on maximal solutions of nonlinear parabolic equations with absorption,, , (2011).   Google Scholar [30] O. Wartlick, A. Kicheva and M. Gonzalez-Gaitan, Morphogen gradient formation,, Cold Spring Harbor Perspectives in Biology, 1 (2009).   Google Scholar [31] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains,, Arch. Rational Mech. Anal., 138 (1997), 279.  doi: 10.1007/s002050050042.  Google Scholar

show all references

##### References:
 [1] G. I. Barenblatt, "Scaling, Self-Similarity, and Intermediate Asymptotics,", Cambridge University Press, (1996).   Google Scholar [2] H. Brézis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions,, J. Math. Pures Appl., 62 (1983), 73.   Google Scholar [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.  doi: 10.1007/BF00251357.  Google Scholar [4] J. Bricmont and A. Kupiainen, Stable non-Gaussian diffusive profiles,, Nonlinear Anal., 26 (1996), 583.  doi: 10.1016/0362-546X(94)00300-7.  Google Scholar [5] Y. Chen and G. Struhl, Dual roles for patched in sequestering and transducing hedgehog,, Cell, 87 (1996), 553.   Google Scholar [6] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar [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.   Google Scholar [8] M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation,, Nonlinear Anal., 11 (1987), 1103.  doi: 10.1016/0362-546X(87)90001-0.  Google Scholar [9] M. Escobedo and O. Kavian, Asymptotic behaviour of positive solutions of a nonlinear heat equation,, Houston J. Math., 14 (1988), 39.   Google Scholar [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.  doi: 10.1080/03605309508821138.  Google Scholar [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.   Google Scholar [12] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).   Google Scholar [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.  doi: 10.1016/0022-0396(84)90042-1.  Google Scholar [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.   Google Scholar [15] L. Herraiz, Asymptotic behaviour of solutions of some semilinear parabolic problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 49.  doi: 10.1016/S0294-1449(99)80008-0.  Google Scholar [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.   Google Scholar [17] S. Kamin and L. A. Peletier, Singular solutions of the heat equation with absorption,, Proc. Amer. Math. Soc., 95 (1985), 205.  doi: 10.2307/2044513.  Google Scholar [18] B. Kawohl, When are solutions to nonlinear elliptic boundary value problems convex?,, Comm. Partial Differential Equations, 10 (1985), 1213.  doi: 10.1080/03605308508820404.  Google Scholar [19] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980).   Google Scholar [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).   Google Scholar [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.  doi: 10.1002/cpa.20014.  Google Scholar [22] A. Martinez-Arias and A. Stewart, "Molecular Principles of Animal Development,", Oxford University Press, (2002).   Google Scholar [23] C. B. Muratov, P. V. Gordon and S. Y. Shvartsman, Self-similar dynamics of morphogen gradients,, Phys. Rev. E, 84 (2011), 1.   Google Scholar [24] L. Oswald, Isolated positive singularities for a nonlinear heat equation,, Houston J. Math., 14 (1988), 543.   Google Scholar [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.  doi: 10.1051/mmnp/20094401.  Google Scholar [26] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).  doi: 10.1007/978-1-4612-5282-5.  Google Scholar [27] G. T. Reeves, C. B. Muratov, T. Schüpbach and S. Y. Shvartsman, Quantitative models of developmental pattern formation,, Devel. Cell, 11 (2006), 289.   Google Scholar [28] G. Sansone, "Equazioni Differenziali nel Campo Reale,", 2. Nicola Zanichelli, 2 (1949).   Google Scholar [29] L. Veron, A note on maximal solutions of nonlinear parabolic equations with absorption,, , (2011).   Google Scholar [30] O. Wartlick, A. Kicheva and M. Gonzalez-Gaitan, Morphogen gradient formation,, Cold Spring Harbor Perspectives in Biology, 1 (2009).   Google Scholar [31] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains,, Arch. Rational Mech. Anal., 138 (1997), 279.  doi: 10.1007/s002050050042.  Google Scholar
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