Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models. (English) Zbl 1411.35157
Summary: This paper deals with the analysis of qualitative properties involved in the dynamics of Keller-Segel type systems in which the diffusion mechanisms of the cells are driven by porous-media flux-saturated phenomena. We study the regularization inside the support of a solution with jump discontinuity at the boundary of the support. We analyze the behavior of the size of the support and blow-up of the solution, and the possible convergence in finite time toward a Dirac mass in terms of the three constants of the system: the mass, the flux-saturated characteristic speed, and the chemoattractant sensitivity constant. These constants of motion also characterize the dynamics of regular and singular traveling waves.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K65 | Degenerate parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C17 | Cell movement (chemotaxis, etc.) |
References:
| [1] | Anderson, A. R. A. and Quaranta, V., Integrative mathematical oncology, Nat. Rev. Cancer8 (2008) 227-234. |
| [2] | Andreu, F., Caselles, V. and Mazón, J. M., The Cauchy problem for a strongly degenerate quasilinear equation, J. Eur. Math. Soc. (JEMS)7 (2005) 361-393. · Zbl 1082.35089 |
| [3] | Andreu, F., Caselles, V., Mazón, J. M. and Moll, S., Finite propagation speed for limited flux diffusion equations, Arch. Ration Mech. Anal.182 (2006) 269-297. · Zbl 1142.35455 |
| [4] | Andreu, F., Caselles, V., Mazón, J. M., Soler, J. and Verbeni, M., Radially symmetric solutions of a tempered diffusion equation. A porous media flux-limited case, SIAM J. Math. Anal.44 (2012) 1019-1049. · Zbl 1276.35109 |
| [5] | Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., Multicellular growing systems: Hyperbolic limits towards macroscopic description, Math. Models Methods Appl. Sci.17 (2007) 1675-1693. · Zbl 1135.92009 |
| [6] | Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., Complexity and mathematical tools toward the modeling of multicellular growing systems, Math. Comput. Modeling51 (2010) 441-451. · Zbl 1190.92001 |
| [7] | Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci.10 (2010) 1179-1207. · Zbl 1402.92065 |
| [8] | Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives, Math. Mod. and Meth. in Appl. Sci.22(1) (2012) 113000137 pp. · Zbl 1328.92023 |
| [9] | Bellomo, N., Bellouquid, A., Tao, Y. and Winkler, M., Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci.25 (2015) 1663-1763. · Zbl 1326.35397 |
| [10] | Bellomo, N. and Winkler, M., Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B4 (2017) 31-67. · Zbl 1367.35044 |
| [11] | Bellomo, N. and Winkler, M., A degenerate chemotaxis system with flux limitation. Maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations42 (2017) 436-473. · Zbl 1430.35166 |
| [12] | Blanchet, A., Dolbeault, J. and Perthame, B., Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations32 (2006) 44. · Zbl 1112.35023 |
| [13] | Bournaveas, N. and Calvez, V., The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity23 (2010) 923. · Zbl 1195.35065 |
| [14] | Calvo, J., Campos, J., Caselles, V., Sánchez, O. and Soler, J., Flux-saturated porous media equation and applications, EMS Surveys Math. Sci.2 (2015) 131-218. · Zbl 1328.35089 |
| [15] | Calvo, J., Campos, J., Caselles, V., Sánchez, O. and Soler, J., Pattern formation in a flux limited reaction-diffusion equation of porous media type, Invent. Math.206 (2016) 57-108. · Zbl 1388.35094 |
| [16] | Calvo, J., Campos, J., Caselles, V., Sánchez, O. and Soler, J., Qualitative behavior for flux-saturated mechanisms: Traveling waves, waiting time and smoothing effects, J. Eur. Math. Soc.19 (2017) 441-472. · Zbl 1379.35167 |
| [17] | Calvo, J., Mazón, J., Soler, J. and Verbeni, M., Qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens, Math. Models Methods Appl. Sci.21 (2011) 893-937. · Zbl 1223.35057 |
| [18] | Campos, J., Guerrero, P., Sánchez, O. and Soler, J., On the analysis of travelling waves to a nonlinear flux limited reaction-diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire30 (2013) 141-155. · Zbl 1263.35059 |
| [19] | Campos, J. and Soler, J., Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Anal.137 (2016) 266-290. · Zbl 1386.35198 |
| [20] | Carrillo, J. A., Caselles, V. and Moll, S., On the relativistic heat equation in one space dimension, Proc. London Math. Soc.107 (2013) 1395-1423. · Zbl 1282.35199 |
| [21] | Caselles, V., Flux limited generalized porous media diffusion equations, Publ. Mat.57 (2013) 144-217. · Zbl 1282.35205 |
| [22] | Caselles, V., Convergence of flux limited porous media diffusion equations to its classical counterpart, Ann. Sc. Norm. Super. Pisa Cl. Sci.14(5) (2015) 481-505. · Zbl 1433.35179 |
| [23] | Chalub, F. A., Dolak-Struss, Y., Markowich, P., Oeltz, D., Schmeiser, C. and Soref, A., Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci.16 (2006) 1173-1198. · Zbl 1094.92009 |
| [24] | Chalub, F. A., Markovich, P., Perthame, B. and Schmeiser, C., Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math.142 (2004) 123-141. · Zbl 1052.92005 |
| [25] | P. H. Chavanis, Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations, preprint (2007), arXiv:0709.1829v1. · Zbl 1189.35335 |
| [26] | Chertock, A., Kurganov, A. and Rosenau, P., Formation of discontinuities in flux-saturated degenerate parabolic equations, Nonlinearity16 (2003) 1875-1898. · Zbl 1049.35111 |
| [27] | Chertock, A., Kurganov, A., Wang, X. and Wu, Y., On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models5 (2012) 51-95. · Zbl 1398.92033 |
| [28] | Maso, G. Dal, Integral representations on \(B V(\Omega)\) of \(\Gamma \)-limits of variational integrals, Manuscripta Math.30 (1980) 387-416. · Zbl 0435.49016 |
| [29] | Dessaud, E., Yang, L. L., Hill, K., Cox, B., Ulloa, F., Ribeiro, A., Mynett, A., Novitch, B. G. and Briscoe, J., Interpretation of the sonic hedgehog morphogen gradient by a temporal adaptation mechanism, Nature450 (2007) 717-720. |
| [30] | Dolbeault, J. and Perthame, B., Optimal critical mass in the two-dimensional Keller-Segel model in \(\mathbb{R}^2\), C. R. Math. Acad. Sci. Paris339 (2004) 611-616. · Zbl 1056.35076 |
| [31] | Dolbeault, J. and Schmeiser, C., The two-dimensional Keller-Segel model after blow-up, Discrete Contin Dyn. Syst.25 (2009) 109-121. · Zbl 1180.35131 |
| [32] | Folkman, J. and Kerbel, K., Clinical translation of angiogenesis inhibitors, Nature Rev. Cancer2 (2002) 727-739. |
| [33] | Giacomelli, L., Finite speed of propagation and waiting-time phenomena for degenerate parabolic equations with linear growth Lagrangian, SIAM J. Math. Anal.47 (2015) 2426-2441. · Zbl 1342.35155 |
| [34] | Gurtin, E. M., MacCamy, R. C. and Socolovsky, E. A., A coordinate transformation for the porous media equation that renders the free boundary stationary, Quart. Appl. Math.42 (1984) 345-357. · Zbl 0574.76093 |
| [35] | Hanahan, D. and Weinberg, R. A., The hallmarks of cancer, Cell100 (2000) 57-70. |
| [36] | Hillen, T. and Painter, K. J., A user’s guide to PDE models for chemotaxis, J. Math. Biol.58 (2009) 183-217. · Zbl 1161.92003 |
| [37] | Hillen, T. and Potapov, A., The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci.27 (2004) 1783-1801. · Zbl 1081.92005 |
| [38] | Keller, E. F., Assessing the Keller-Segel model: How has it fared? In Biological Growth and Spread, Proc. Conf., Heidelberg, , 1979 (Springer, 1980), pp. 379-387. · Zbl 0437.92003 |
| [39] | Keller, E. F. and Segel, L. A., Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol.30 (1971) 235-248. · Zbl 1170.92308 |
| [40] | Kondo, S. and Miura, T., Reaction-Diffusion model as a framework for understanding biological pattern formation, Science329 (2010) 1616-1620. · Zbl 1226.35077 |
| [41] | Kurganov, A. and Rosenau, P., On reaction processes with saturating diffusion, Nonlinearity19 (2006) 171-193. · Zbl 1094.35063 |
| [42] | Lachowicz, M., Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors, Math. Models Methods Appl. Sci.15 (2005) 1667-1683. · Zbl 1078.92036 |
| [43] | O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 (Amer. Math. Soc., 1968). · Zbl 0174.15403 |
| [44] | Levermore, C. D. and Pomraning, G. C., A flux-limited diffusion theory, Astrophys. J.248 (1981) 321-334. |
| [45] | Lieberman, G. M., Second Order Parabolic Differential Equations (World Scientific, 2005). |
| [46] | Meirmanov, A. M., Pukhnachov, V. V. and Shmarev, S. I., Evolution Equations and Lagrangian Coordinates, , Vol. 24 (Walter de Gruyter, 1997). · Zbl 0876.35001 |
| [47] | Mihalas, D. and Mihalas, B., Foundations of Radiation Hydrodynamics (Oxford Univ. Press, 1984). · Zbl 0651.76005 |
| [48] | Othmer, H. and Hillen, T., The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math.62 (2002) 1222-1250. · Zbl 1103.35098 |
| [49] | Patlak, C. S., Random walk with persistence and external bias, Bull. Math. Biophys.15 (1953) 311-338. · Zbl 1296.82044 |
| [50] | Rosenau, P., Tempered diffusion: A transport process with propagating front and inertial delay, Phys. Rev. A46 (1992) 7371-7374. |
| [51] | Verbeni, M., Sánchez, O., Mollica, E., Siegl-Cachedenier, I., Carleton, A., Guerrero, I., Ruiz i Altaba, A. and Soler, J., Modeling morphogenetic action through flux-limited spreading, Phys. Life Rev.10 (2013) 457-475. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
