Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. (English) Zbl 1326.35397
The paper is an overview supplemented with a critical analysis on the qualitative study of mathematical problems for the original Keller-Segel system of chemotaxis as well as on a variety of models derived with the aim of improving its consistency with biological reality. The first part discusses models including the classical Keller-Segel system, systems with a logistic growth term, chemotaxis with signal-dependent sensitivities, nonlinear cell diffusion model and chemotaxis-fluid systems. The second part is devoted to the qualitative analysis of analytic questions such as the existence of solutions, their asymptotic behavior and blow-up. The third part deals with the derivation of macroscopic models by means of kinetic theory methods. Several open problems are formulated as suggestions for future research activities.
Reviewer: Piotr Biler (Wroclaw)
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 35K55 | Nonlinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
References:
| [1] | Aida, M., J. London Math. Soc., 74, 453, 2006 |
| [2] | Alber, M., Phys. Rev. Lett., 99, 168102, 2007 |
| [3] | Allen, B.; Nowak, M., EMS Surv. Math. Sci., 1, 113, 2014 · Zbl 1303.91040 |
| [4] | Anderson, A. R. A.; Chaplain, M. A. J., Bull. Math. Biol., 60, 857, 1998 · Zbl 0923.92011 |
| [5] | Anderson, A. R. A., J. Theoret. Medicine, 2, 129, 2000 · Zbl 0947.92012 |
| [6] | Andreu, F.; Caselles, V.; Mazón, J. M., Arch. Rational Mech. Anal., 176, 415, 2005 · Zbl 1112.35111 |
| [7] | Andreu, F., Arch. Rational Mech. Anal., 182, 269, 2006 · Zbl 1142.35455 |
| [8] | Banasiak, J.; Lachowicz, M., Methods of Small Parameter in Mathematical Bio-logy, 2014, Birkhäuser · Zbl 1309.92012 |
| [9] | Bellomo, N.; Bellouquid, A., Nonlinear Anal. Hybrid Syst., 3, 215, 2009 · Zbl 1184.93071 |
| [10] | Bellomo, N.; Bellouquid, A., Front. Sci. Eng., 4, 1, 2014 |
| [11] | Bellomo, N.; Bellouquid, A., Math. Mech. Solids, 20, 268, 2015 · Zbl 1327.76175 |
| [12] | Bellomo, N.Bellouquid, A., Commun. Math. Sci.13(39), (2015). |
| [13] | Bellomo, N., Math. Models Methods Appl. Sci., 17, 1675, 2007 · Zbl 1135.92009 |
| [14] | Bellomo, N., Math. Models Methods Appl. Sci., 20, 1179, 2010 · Zbl 1402.92065 |
| [15] | Bellomo, N., Math. Comput. Model., 51, 441, 2010 · Zbl 1190.92001 |
| [16] | Bellomo, N., Math. Models Methods Appl. Sci., 22, 1130001, 2012 · Zbl 1328.92023 |
| [17] | Bellomo, N., Discrete Contin. Dynam. Syst. Ser. B, 18, 847, 2013 |
| [18] | Bellomo, N., Discrete Contin. Dynam. Syst. Ser. B, 19, 1869, 2014 |
| [19] | Bellomo, N.; Knopoff, D.; Soler, J., Math. Models Methods Appl. Sci., 23, 1861, 2013 · Zbl 1315.35137 |
| [20] | Bellomo, N.; Piccoli, B.; Tosin, A., Math. Models Methods Appl. Sci., 22, 1230004, 2012 · Zbl 1252.35192 |
| [21] | Bellouquid, A., Math. Models Methods Appl. Sci., 14, 853, 2004 · Zbl 1076.76066 |
| [22] | Bellouquid, A.; De Angelis, E., Nonlinear Anal. Real World Appl., 12, 1101, 2011 |
| [23] | Bellouquid, A.; De Angelis, E.; Knopoff, D., Math. Models Methods Appl. Sci., 23, 949, 2013 · Zbl 1303.92040 |
| [24] | Bellouquid, A.; Delitala, M., Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, 2006, Birkhäuser · Zbl 1178.92002 |
| [25] | Bellouquid, A.; Delitala, M., Math. Models Methods Appl. Sci., 15, 1639, 2005 · Zbl 1093.82016 |
| [26] | Biler, P., Adv. Math. Sci. Appl., 9, 347, 1999 · Zbl 0941.35009 |
| [27] | Biler, P.; Corrias, L.; Dolbeault, J., J. Math. Biol., 63, 1, 2011 · Zbl 1230.92011 |
| [28] | Biler, P.; Hebisch, W.; Nadzieja, T., Nonlinear Anal., 23, 1189, 1994 · Zbl 0814.35054 |
| [29] | Biler, P.; Karch, G., J. Evol. Equations, 10, 247, 2010 · Zbl 1239.35177 |
| [30] | Blanchet, A.; Carrillo, J. A.; Laurençot, Ph., Calc. Var. Partial Differential Equations, 35, 133, 2009 · Zbl 1172.35035 |
| [31] | Brenier, Y., Optimal Transportation and Applications, 1813 (Springer, 2003) pp. 91-122. |
| [32] | Brenier, Y., J. Nonlinear Sci., 19, 547, 2009 · Zbl 1177.49064 |
| [33] | Borsche, R., Math. Models Methods Appl. Sci., 24, 221, 2014 · Zbl 1321.92042 |
| [34] | Borsche, R., Math. Models Methods Appl. Sci., 24, 359, 2014 · Zbl 1279.90035 |
| [35] | Burczak, J.; Cieślak, T.; Morales Rodrigo, C., Nonlinear Anal., 75, 5215, 2012 |
| [36] | Burger, M.; Di Francesco, M.; Dolak-Struss, Y., SIAM J. Math. Anal., 38, 1288, 2006 · Zbl 1114.92008 |
| [37] | Calvez, V.; Carrillo, J. A., J. Math. Pures Appl., 86, 155, 2006 · Zbl 1116.35057 |
| [38] | Cao, X., J. Math. Anal. Appl., 412, 181, 2014 · Zbl 1364.35123 |
| [39] | Cao, X., Discrete Contin. Dynam. Syst. Ser. A, 35, 1891, 2015 · Zbl 1515.35047 |
| [40] | X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis system, preprint. |
| [41] | Cao, X.; Ishida, S., Nonlinearity, 27, 1899, 2014 · Zbl 1301.35056 |
| [42] | Cao, X.; Zheng, S., Math. Models Methods Appl. Sci., 37, 2326, 2014 · Zbl 1370.92029 |
| [43] | Cerreti, F., Math. Models Methods Appl. Sci., 21, 825, 2011 · Zbl 1221.35088 |
| [44] | Chae, M.; Kang, K.; Lee, J., Discrete Contin. Dynam. Syst. Ser. A, 33, 2271, 2013 · Zbl 1277.35276 |
| [45] | Chae, M.; Kang, K.; Lee, J., Comm. Partial Differential Equations, 39, 1205, 2014 · Zbl 1304.35481 |
| [46] | Chalub, F. A., Math. Models Methods Appl. Sci., 16, 1173, 2006 · Zbl 1094.92009 |
| [47] | Chalub, F. A.; Kang, K., Nonlinear Anal., 64, 686, 2006 · Zbl 1103.35046 |
| [48] | Chalub, F. A., Monatsh. Math., 142, 123, 2004 |
| [49] | Chaplain, M. A. J.; Lolas, G., Math. Models Methods Appl. Sci., 18, 1685, 2005 |
| [50] | Chaplain, M. A. J.; Lolas, G., Netw. Heterog. Media, 1, 399, 2006 · Zbl 1108.92023 |
| [51] | Chen, K. C.; Ford, R. M.; Cummings, P. T., SIAM J. Appl. Math., 59, 35, 1999 · Zbl 0918.60023 |
| [52] | Chertock, A., J. Fluid Mech., 694, 155, 2012 · Zbl 1250.76191 |
| [53] | Chertock, A., Kinet. Relat. Models, 5, 51, 2012 · Zbl 1398.92033 |
| [54] | Cieślak, T.; Laurençot, Ph., Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 437, 2010 · Zbl 1270.35377 |
| [55] | Cieślak, T.; Stinner, C., J. Differential Equations, 252, 5832, 2012 · Zbl 1252.35087 |
| [56] | Cieślak, T.; Stinner, C., Acta Appl. Math., 129, 135, 2014 · Zbl 1295.35123 |
| [57] | Cieślak, T.; Stinner, C., J. Differential Equations, 258, 2080, 2015 · Zbl 1331.35041 |
| [58] | Coll, J. C., Mar. Biol., 118, 177, 1994 |
| [59] | Corrias, L.; Perthame, B., Math. Comput. Model., 47, 755, 2008 · Zbl 1134.92006 |
| [60] | Corrias, L.; Perthame, B.; Zaag, H., C. R. Acad. Sci. Paris, Ser. I, 336, 141, 2003 · Zbl 1028.35062 |
| [61] | Corrias, L.; Perthame, B.; Zaag, H., Milan J. Math., 72, 1, 2004 · Zbl 1115.35136 |
| [62] | D’Acunto, B.; Frunzo, L., Math. Comput. Model., 53, 1596, 2011 · Zbl 1219.35324 |
| [63] | De Angelis, E., Math. Models Methods Appl. Sci., 24, 2723, 2014 · Zbl 1328.92050 |
| [64] | M. del Pino, G. Vaira and A. Pistoia, Large mass boundary condensation patterns in the stationary Keller-Segel system, arXiv: 1403.2511. |
| [65] | del Pino, M.; Wei, J., Nonlinearity, 19, 661, 2006 · Zbl 1137.35007 |
| [66] | Dessaud, E., Nature, 450, 717, 2007 |
| [67] | DiFrancesco, M.; Lorz, A.; Markowich, P. A., Discrete Contin. Dynam. Syst. Ser. A, 28, 1437, 2010 · Zbl 1276.35103 |
| [68] | Dolak, Y.; Schmeiser, C., J. Math. Biol., 51, 595, 2005 · Zbl 1077.92003 |
| [69] | Dolbeault, J.; Schmeiser, C., Discrete Contin. Dynam. Syst. Ser. A, 25, 109, 2009 · Zbl 1180.35131 |
| [70] | Dombowski, C., Phys. Rev. Lett., 93, 098103, 2004 |
| [71] | Duan, R. J.; Lorz, A.; Markowich, P. A., Comm. Partial Differential Equations, 35, 1635, 2010 · Zbl 1275.35005 |
| [72] | Duan, R.; Xiang, Z., Int. Math. Res. Notices, 2012, 20, 2012 |
| [73] | Efendiev, M.; Nakaguchi, E., Adv. Math. Sci. Appl., 16, 569, 2006 · Zbl 1130.37402 |
| [74] | Espejo, E.; Suzuki, T., Nonlinear Anal. Real World Appl., 21, 110, 2015 · Zbl 1302.35102 |
| [75] | Feireisl, E.; Laurencõt, P.; Petzeltovo, H., J. Differential Equations, 236, 551, 2010 |
| [76] | Filbet, F.; Laurençot, P.; Perthame, B., J. Math. Biol., 50, 189, 2005 · Zbl 1080.92014 |
| [77] | Folkman, J., Semin. Oncol., 29, 15, 2002 |
| [78] | Folkman, J.; Kerbel, K., Nature Rev. Cancer, 2, 727, 2002 |
| [79] | Fontelos, M. A.; Friedman, A.; Hu, B., SIAM J. Math. Anal., 33, 1330, 2002 · Zbl 1020.35030 |
| [80] | Friedman, A.; McLeod, B., Indiana Univ. Math. J., 34, 425, 1985 · Zbl 0576.35068 |
| [81] | Fujie, K., J. Math. Anal. Appl., 424, 675, 2015 · Zbl 1310.35144 |
| [82] | K. Fujie, A. Itô, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, preprint. · Zbl 1322.35059 |
| [83] | K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, preprint. · Zbl 1330.35051 |
| [84] | Fujie, K.; Winkler, M.; Yokota, T., Math. Models Methods Appl. Sci., 38, 1212, 2014 |
| [85] | Fujie, K.; Winkler, M.; Yokota, T., Nonlinear Anal., 109, 56, 2014 · Zbl 1297.35051 |
| [86] | Gajewski, H.; Zacharias, K., Math. Nachr., 195, 77, 1998 · Zbl 0918.35064 |
| [87] | Gerisch, A.; Painter, K. J.; Chauviere, A.; Preziosi, L.; Verdier, C., Chapter 12: Mathematical Modeling of Cell Adhesion and Its Applications to Developmental Biology and Cancer Invasion, 2010, Chapman & Hall/CRC |
| [88] | Giga, Y.; Kohn, R. V., Comm. Pure Appl. Math., 38, 297, 1985 · Zbl 0585.35051 |
| [89] | Giga, Y.; Sohr, H., J. Funct. Anal., 102, 72, 1991 · Zbl 0739.35067 |
| [90] | Harada, G., Adv. Differential Equations, 6, 1255, 2001 |
| [91] | Haskovec, J.; Schmeiser, C., J. Statist. Phys., 135, 133, 2009 · Zbl 1173.82021 |
| [92] | Haskovec, J.; Schmeiser, C., Comm. Partial Differential Equations, 36, 940, 2011 · Zbl 1229.35111 |
| [93] | Herrero, M. A.; Köhn, A.; Pérez-Pomares, J. M., Math. Models Methods Appl. Sci., 19, 1483, 2009 · Zbl 1180.35286 |
| [94] | Herrero, M. A.; Velázquez, J. J. L., Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24, 633, 1997 · Zbl 0904.35037 |
| [95] | Hill, N. A.; Pedley, T. J., Fluid Dynam. Res., 37, 1, 2005 · Zbl 1153.76455 |
| [96] | Hillen, T., Discrete Contin. Dynam. Syst. Ser. B, 4, 961, 2004 · Zbl 1052.92006 |
| [97] | Hillen, T.; Othmer, H., SIAM J. Appl. Math., 61, 751, 2000 · Zbl 1002.35120 |
| [98] | Hillen, T.; Painter, K. J., J. Math. Biol., 58, 183, 2009 · Zbl 1161.92003 |
| [99] | Hillen, T.; Painter, K.; Winkler, M., Math. Models Methods Appl. Sci., 23, 165, 2013 · Zbl 1263.35204 |
| [100] | Horstmann, D., Jahresberichte Deutsch. Math.-Verein, 105, 103, 2003 · Zbl 1071.35001 |
| [101] | Horstmann, D.; Wang, G., Eur. J. Appl. Math., 12, 159, 2001 · Zbl 1017.92006 |
| [102] | Horstmann, D.; Winkler, M., J. Differential Equations, 215, 52, 2005 · Zbl 1085.35065 |
| [103] | Ishida, S.; Seki, K.; Yokota, T., J. Differential Equations, 256, 2993, 2014 · Zbl 1295.35252 |
| [104] | Ishida, S.; Yokota, T., J. Differential Equations, 252, 1421, 2012 · Zbl 1237.35093 |
| [105] | Ishida, S.; Yokota, T., J. Differential Equations, 252, 2469, 2012 · Zbl 1241.35118 |
| [106] | Kang, K.; Stevens, A.; Velázquez, J. J. L., Comm. Partial Differential Equations, 35, 245, 2010 · Zbl 1193.35013 |
| [107] | Kavallaris, N.; Souplet, Ph., SIAM J. Math. Anal., 41, 128, 2009 |
| [108] | Keller, E. F., Biological Growth and Spread (Springer, 1980) pp. 379-387. |
| [109] | Keller, E. F.; Segel, L. A., J. Theor. Biol., 26, 399, 1970 · Zbl 1170.92306 |
| [110] | Keller, E. F.; Segel, L. A., J. Theor. Biol., 30, 225, 1971 · Zbl 1170.92307 |
| [111] | Kiselev, A.; Ryzhik, L., Comm. Partial Differential Equations, 37, 298, 2012 · Zbl 1236.35190 |
| [112] | Klapper, I.; Dockeryt, J., SIAM Rev., 52, 221, 2010 · Zbl 1191.92065 |
| [113] | Kond, S.; Miura, T., Science, 329, 1616, 2010 |
| [114] | Kowalczyk, R., J. Math. Anal. Appl., 305, 566, 2005 · Zbl 1065.35063 |
| [115] | Kozono, H.; Sugiyama, Y., J. Differential Equations, 247, 1, 2009 · Zbl 1207.35178 |
| [116] | H. Kozono and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, preprint. · Zbl 1343.35069 |
| [117] | Kuto, K., Physica D, 241, 1629, 2012 |
| [118] | Lachowicz, M., Multiscale Problems in the Life Sciences, 1940 (Springer, 2008) pp. 201-267. · Zbl 1264.92050 |
| [119] | Lankeit, J., J. Differential Equations, 258, 1158, 2015 · Zbl 1319.35085 |
| [120] | J. Lankeit, Chemotaxis can prevent thresholds on population density, to appear in Discrete Contin. Dynam. Syst. Ser. B. |
| [121] | J. Lankeit, A new approach toward global existence in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Models Methods Appl. Sci., DOI: 10.1002/mma.3489. |
| [122] | Levermore, C. D., J. Stat. Phys., 83, 1021, 1996 · Zbl 1081.82619 |
| [123] | Li, T., Math. Models Methods Appl. Sci., 25, 721, 2015 |
| [124] | Li, T.; Wang, Z.-A., J. Differential Equations, 250, 1310, 2011 · Zbl 1213.35109 |
| [125] | Lions, P. L., Arch. Rational Mech. Anal., 74, 335, 1980 · Zbl 0449.35036 |
| [126] | Liţcanu, G.; Morales-Rodrigo, C., Math. Models Methods Appl. Sci., 20, 1721, 2010 · Zbl 1213.35250 |
| [127] | Liu, J.-G.; Lorz, A., Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 643, 2011 · Zbl 1236.92013 |
| [128] | Lorenz, T.; Surulescu, C., Math. Models Methods Appl. Sci., 24, 2383, 2014 · Zbl 1300.92016 |
| [129] | Luckhaus, S.; Sugiyama, Y., Indiana Univ. Math. J., 56, 1279, 2007 · Zbl 1118.35006 |
| [130] | Luckhaus, S.; Sugiyama, Y.; Velázquez, J. J. L., Arch. Rational Mech. Anal., 206, 31, 2012 · Zbl 1256.35180 |
| [131] | Maini, P. K., J. Math. Biol., 27, 507, 1989 · Zbl 0716.92004 |
| [132] | Mallet, D. G.; Pettet, G. J., Bull. Math. Biol., 68, 231, 2006 · Zbl 1334.92061 |
| [133] | Manásevich, R.; Phan, Q. H.; Souplet, Ph., Eur. J. Appl. Math., 24, 273, 2013 · Zbl 1284.35445 |
| [134] | Marciniak-Czochra, A.; Ptashnyk, M., Math. Models Methods Appl. Sci., 20, 449, 2010 · Zbl 1194.35043 |
| [135] | Miller, R. L., J. Exp. Zool., 234, 383, 1985 |
| [136] | Mizoguchi, N., Calc. Var. Partial Differential Equations, 48, 491, 2013 · Zbl 1288.35286 |
| [137] | Mizoguchi, N.; Souplet, Ph., Ann. Inst. H. Poincaré Anal. Non Linéaire, 31, 851, 2014 · Zbl 1302.35075 |
| [138] | N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. |
| [139] | N. Mizoguchi and M. Winkler, Boundedness of global solutions in the two-dimensional parabolic Keller-Segel system, preprint. |
| [140] | Muller, I.; Ruggeri, T., Rational Extended Thermodynamics, 1998, Springer · Zbl 0895.00005 |
| [141] | Murray, J. D., Mathematical Biology II. Spatial Models and Biomedical Applications, 2003, Springer · Zbl 1006.92002 |
| [142] | Musso, M.; Wei, J., Asympt. Anal., 49, 217, 2006 · Zbl 1115.35051 |
| [143] | Nagai, T.; Senba, T., Adv. Math. Sci. Appl., 8, 145, 1998 · Zbl 0902.35010 |
| [144] | Nagai, T.; Senba, T.; Suzuki, T., Hiroshima Math. J., 30, 463, 2000 · Zbl 0984.35079 |
| [145] | Nagai, T.; Senba, T.; Yoshida, K., Funkcial. Ekvac., 40, 411, 1997 · Zbl 0901.35104 |
| [146] | Nagai, T.; Senba, T.; Yoshida, K., RIMS Kokyuroku, 1009, 22, 1997 · Zbl 0931.92005 |
| [147] | Nakaguchi, E.; Osaki, K., Nonlinear Anal., 74, 286, 2011 |
| [148] | Nakaguchi, E.; Osaki, K., Discrete Contin. Dynam. Syst. Ser. B, 18, 2627, 2013 · Zbl 1318.37028 |
| [149] | E. Nakaguchi and K. Osaki, L p -estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, to appear in Funkcial. Ekvac. · Zbl 1345.35121 |
| [150] | Nowak, M. A., Evolutionary Dynamics — Exploring the Equations of Life, 2006, Harvard Univ. Press · Zbl 1115.92047 |
| [151] | Nowak, M. A.; Sigmund, K., Science, 303, 793, 2004 |
| [152] | Olsen, L., Math. Biosci., 158, 145, 1999 |
| [153] | Olsen, L.; Sherratt, J. A.; Maini, P. K., Bull. Math. Biol., 58, 787, 1996 · Zbl 0865.92010 |
| [154] | Osaki, K., Nonlinear Anal., 51, 119, 2002 |
| [155] | Osaki, K.; Yagi, A., Funkcial. Ekvac., 44, 441, 2001 · Zbl 1145.37337 |
| [156] | Othmer, H. G.; Dunbar, S. R.; Alt, W., J. Math. Biol., 26, 263, 1988 · Zbl 0713.92018 |
| [157] | Othmer, H. G.; Hillen, T., SIAM J. Appl. Math., 62, 1222, 2002 · Zbl 1103.35098 |
| [158] | Painter, K. J.; Hillen, T., Canad. Appl. Math. Quart., 10, 501, 2002 · Zbl 1057.92013 |
| [159] | Painter, K. J.; Hillen, T., Physica D, 240, 363, 2011 · Zbl 1255.37026 |
| [160] | Painter, K. J.; Maini, P. K.; Othmer, H. G., J. Math. Biol., 41, 285, 2000 · Zbl 0969.92003 |
| [161] | Patlak, C. S., Bull. Math. Biol., 15, 311, 1953 · Zbl 1296.82044 |
| [162] | Payne, L. E.; Straughan, B., Studies Appl. Math., 123, 337, 2009 · Zbl 1180.35529 |
| [163] | Perthame, B., Appl. Math., 49, 539, 2004 · Zbl 1099.35157 |
| [164] | Perthame, B.; Dalibard, A. L., Trans. Amer. Math. Soc., 361, 2319, 2009 · Zbl 1180.35343 |
| [165] | Quittner, P.; Souplet, Ph., Superlinear Parabolic Problems. Blowup, Global Existence and Steady States, 2007, Birkhäuser · Zbl 1128.35003 |
| [166] | Ringhofer, C.; Schmeiser, C.; Zwirchmayr, A., SIAM J. Numer. Anal., 39, 1078, 2001 · Zbl 0994.82080 |
| [167] | Roth, S., Dev. Genes Evol., 221, 255, 2011 |
| [168] | Saragosti, J., PLoS Comput. Biol., 6, 1000890, 2011 |
| [169] | R. Schweyer, Stable blow-up dynamic for the parabolic-parabolic Patlak-Keller-Segel model, arXiv: 1403.4975. |
| [170] | Senba, T.; Suzuki, T., Adv. Math. Sci. Appl., 10, 191, 2000 · Zbl 0999.35031 |
| [171] | Senba, T.; Suzuki, T., Abstr. Appl. Anal., 2006, 1, 2006 |
| [172] | Sohr, H., The Navier-Stokes Equations. An Elementary Functional Analytic Approach, 2001, Birkhäuser · Zbl 0983.35004 |
| [173] | Stolarska, M. A.; Yangjin, K. I. M.; Othmer, H., Philos. Trans. A Math. Phys. Eng. Sci., 67, 3526, 2009 |
| [174] | Stinner, C.; Surulescu, C.; Winkler, M., SIAM J. Math. Anal., 46, 1969, 2014 · Zbl 1301.35189 |
| [175] | Stinner, C.; Winkler, M., Nonlinear Anal. Real World Appl., 12, 3727, 2011 · Zbl 1268.35072 |
| [176] | Sugiyama, Y., Differential Integral Equations, 20, 133, 2007 · Zbl 1212.35241 |
| [177] | Sugiyama, Y., Math. Nachr., 285, 744, 2012 · Zbl 1276.35102 |
| [178] | Sugiyama, Y.; Kunii, H., J. Differential Equations, 227, 333, 2006 · Zbl 1102.35046 |
| [179] | Tao, Y., J. Math. Anal. Appl., 354, 60, 2009 · Zbl 1160.92027 |
| [180] | Tao, Y., Nonlinear Anal. Real World Appl., 12, 418, 2011 · Zbl 1205.35144 |
| [181] | Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407-7382. |
| [182] | Tao, Y.; Wang, M., SIAM J. Math. Anal., 41, 1533, 2009 · Zbl 1201.35005 |
| [183] | Tao, Y.; Wang, Z.-A., Math. Models Methods Appl. Sci., 23, 1, 2013 · Zbl 1403.35136 |
| [184] | Tao, Y.; Wang, L.; Wang, Z. A., Discrete Contin. Dynam. Syst. Ser. B, 18, 821, 2013 · Zbl 1272.35040 |
| [185] | Tao, Y.; Winkler, M., SIAM J. Math. Anal., 43, 685, 2011 · Zbl 1259.35210 |
| [186] | Tao, Y.; Winkler, M., J. Differential Equations, 252, 692, 2012 · Zbl 1382.35127 |
| [187] | Tao, Y.; Winkler, M., Discrete Contin. Dynam. Syst. Ser. A, 32, 1901, 2012 · Zbl 1276.35105 |
| [188] | Tao, Y.; Winkler, M., J. Differential Equations, 252, 2520, 2012 · Zbl 1268.35016 |
| [189] | Tao, Y.; Winkler, M., Ann. Inst. Henri Poincaré Anal. Non Linéaire, 30, 157, 2013 · Zbl 1283.35154 |
| [190] | Tao, Y.; Winkler, M., J. Differential Equations, 257, 784, 2014 · Zbl 1295.35144 |
| [191] | Tao, Y.; Winkler, M., Nonlinearity, 27, 1225, 2014 · Zbl 1293.35047 |
| [192] | Tao, Y.; Winkler, M., Proc. Roy. Soc. Edinburgh Sect. A, 144, 1067, 2014 · Zbl 1312.35171 |
| [193] | Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, preprint. · Zbl 1328.35103 |
| [194] | Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, preprint. · Zbl 1356.35054 |
| [195] | Tello, J. I.; Winkler, M., Comm. Partial Differential Equations, 32, 849, 2007 · Zbl 1121.37068 |
| [196] | Tello, J. I.; Winkler, M., Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12, 833, 2013 · Zbl 1295.35136 |
| [197] | Tindall, M. J., Bull. Math. Biol., 70, 1570, 2008 |
| [198] | Tuval, I., Proc. Natl. Acad. Sci. USA, 102, 2277, 2005 |
| [199] | Vázquez, J. L., The Porous Medium Equation. Mathematical Theory, 2006, Oxford Univ. Press |
| [200] | Verbeni, M., Phys. Life Rev., 10, 457, 2013 |
| [201] | Vorotnikov, D., Commun. Math. Sci., 12, 545, 2014 · Zbl 1302.35069 |
| [202] | Walker, C.; Webb, G. F., SIAM J. Math. Anal., 38, 1694, 2007 · Zbl 1120.92024 |
| [203] | L. Wang and Ch. Mu, Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, to appear in Z. Angew. Math. Phys., DOI: 10.1007/s00033-014-0491-9. · Zbl 1328.92019 |
| [204] | Wang, L.; Mu, Ch.; Zheng, P., J. Differential Equations, 256, 1847, 2014 · Zbl 1301.35060 |
| [205] | Wang, L.; Mu, Ch.; Zhou, Sh., Z. Angew. Math. Phys., 65, 1137, 2014 · Zbl 1305.92024 |
| [206] | Y. Wang and X. Cao, Global classical solutions of a 3D chemotaxis-Stokes system with rotation, to appear in Discrete Contin. Dynam. Syst. Ser. B. |
| [207] | Wang, Z. A.; Winkler, M.; Wrzosek, D., Nonlinearity, 24, 3279, 2011 · Zbl 1255.35213 |
| [208] | Wang, Z. A.; Winkler, M.; Wrzosek, D., SIAM J. Math. Anal., 44, 3502, 2012 · Zbl 1277.35223 |
| [209] | Winkler, M., J. Math. Anal. Appl., 348, 708, 2008 · Zbl 1147.92005 |
| [210] | Winkler, M., Comm. Partial Differential Equations, 35, 1516, 2010 · Zbl 1290.35139 |
| [211] | Winkler, M., J. Differential Equations, 248, 2889, 2010 · Zbl 1190.92004 |
| [212] | Winkler, M., Math. Models Methods Appl. Sci., 33, 12, 2010 · Zbl 1182.35220 |
| [213] | Winkler, M., J. Math. Anal. Appl., 384, 261, 2011 · Zbl 1241.35028 |
| [214] | Winkler, M., Math. Models Methods Appl. Sci., 34, 176, 2011 · Zbl 1291.92018 |
| [215] | Winkler, M., Math. Nachr., 283, 1664, 2011 |
| [216] | Winkler, M., Comm. Partial Differential Equations, 37, 319, 2012 · Zbl 1236.35192 |
| [217] | Winkler, M., J. Math. Pures Appl., 100, 748, 2013 · Zbl 1326.35053 |
| [218] | Winkler, M., Arch. Rational Mech. Anal., 211, 455, 2014 · Zbl 1293.35220 |
| [219] | Winkler, M., J. Differential Equations, 257, 1056, 2014 · Zbl 1293.35048 |
| [220] | Winkler, M., J. Nonlinear Sci., 24, 809, 2014 · Zbl 1311.35040 |
| [221] | M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. Henri Poincaré Anal. Non Linéaire, to appear. · Zbl 1351.35239 |
| [222] | M. Winkler, Boundedness and large time behavior in a three-dimensional chemo-taxis-Stokes system with nonlinear diffusion and general sensitivity, arXiv: 1501. 07059. |
| [223] | M. Winkler, A two-dimensional chemotaxis-Stokes system with rotational flux: Global solvability, eventual smoothness and stabilization, preprint. |
| [224] | M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, preprint. · Zbl 1330.35202 |
| [225] | Wrzosek, D., Proc. Roy. Soc. Edinburgh Sect. A, 136, 431, 2006 · Zbl 1104.35007 |
| [226] | Wrzosek, D., Math. Model. Nat. Phenom., 5, 123, 2010 · Zbl 1184.35166 |
| [227] | Xue, C., J. Math. Biol., 70, 1, 2015 · Zbl 1375.92012 |
| [228] | Xue, C.; Othmer, H. G., SIAM J. Appl. Math., 70, 133, 2009 · Zbl 1184.35308 |
| [229] | Yang, Y.; Chen, H.; Liu, W., SIAM J. Math. Anal., 33, 763, 2001 · Zbl 1029.35132 |
| [230] | Zhang, Q.; Zheng, X., SIAM J. Math. Anal., 46, 3078, 2014 · Zbl 1444.35011 |
| [231] | Zheng, P., J. Math. Anal. Appl., 424, 509, 2015 |
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