Critical mass on the Keller-Segel system with signal-dependent motility

Jin, Hai-Yang; Wang, Zhi-An

doi:10.1090/proc/15124

Critical mass on the Keller-Segel system with signal-dependent motility
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by Hai-Yang Jin and Zhi-An Wang PDF

Proc. Amer. Math. Soc. 148 (2020), 4855-4873 Request permission

Abstract:

This paper is concerned with the global boundedness and blow-up of solutions to the Keller-Segel system with density-dependent motility in a two-dimensional bounded smooth domain with Neumman boundary conditions. We show that if the motility function decays exponentially, then a critical mass phenomenon similar to the minimal Keller-Segel model will arise. That is, there is a number $m_*>0$, such that the solution will globally exist with uniform-in-time bound if the initial cell mass (i.e., $L^1$-norm of the initial value of cell density) is less than $m_*$, while the solution may blow up if the initial cell mass is greater than $m_*$.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
Jaewook Ahn and Changwook Yoon, Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing, Nonlinearity 32 (2019), no. 4, 1327–1351. MR 3923170, DOI 10.1088/1361-6544/aaf513
N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations 4 (1979), no. 8, 827–868. MR 537465, DOI 10.1080/03605307908820113
Herbert Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations 3 (1990), no. 1, 13–75. MR 1014726
Herbert Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993, pp. 9–126. MR 1242579, DOI 10.1007/978-3-663-11336-2_{1}
N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), no. 9, 1663–1763. MR 3351175, DOI 10.1142/S021820251550044X
Piotr Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl. 9 (1999), no. 1, 347–359. MR 1690388
Adrien Blanchet and Philippe Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion, Commun. Pure Appl. Anal. 11 (2012), no. 1, 47–60. MR 2833337, DOI 10.3934/cpaa.2012.11.47
D. Burini and N. Chouhad, A multiscale view of nonlinear diffusion in biology: from cells to tissues, Math. Models Methods Appl. Sci. 29 (2019), no. 4, 791–823. MR 3945376, DOI 10.1142/S0218202519400062
X. Fu, L. H. Tang, C. Liu, J. D. Huang, T. Hwa, and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108:198102, 2012.
Kentarou Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl. 424 (2015), no. 1, 675–684. MR 3286587, DOI 10.1016/j.jmaa.2014.11.045
Kentarou Fujie, Michael Winkler, and Tomomi Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci. 38 (2015), no. 6, 1212–1224. MR 3338145, DOI 10.1002/mma.3149
Kentarou Fujie and Takasi Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 1, 81–102. MR 3426833, DOI 10.3934/dcdsb.2016.21.81
Kentarou Fujie and Takasi Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity 29 (2016), no. 8, 2417–2450. MR 3538418, DOI 10.1088/0951-7715/29/8/2417
T. Hillen and K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), no. 1-2, 183–217. MR 2448428, DOI 10.1007/s00285-008-0201-3
Dirk Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (2003), no. 3, 103–165. MR 2013508
Dirk Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein. 106 (2004), no. 2, 51–69. MR 2073515
Dirk Horstmann and Guofang Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math. 12 (2001), no. 2, 159–177. MR 1931303, DOI 10.1017/S0956792501004363
Dirk Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 4, 399–423. MR 1867320, DOI 10.1007/PL00001455
Kazuhiro Ishige, Philippe Laurençot, and Noriko Mizoguchi, Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller-Segel system, Math. Ann. 367 (2017), no. 1-2, 461–499. MR 3606447, DOI 10.1007/s00208-016-1400-7
Hai-Yang Jin, Yong-Jung Kim, and Zhi-An Wang, Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math. 78 (2018), no. 3, 1632–1657. MR 3814035, DOI 10.1137/17M1144647
Hai-Yang Jin and Zhi-An Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations 260 (2016), no. 1, 162–196. MR 3411669, DOI 10.1016/j.jde.2015.08.040
Hai-Yang Jin and Zhi-An Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, Euro. J. Appl. Math., accpeted, 2020.
P. Kareiva and G. Odell, Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Amer. Nat., 130(2):233-270, 1987.
E. F. Keller and L. A. Segel, Models for chemtoaxis, J. Theor. Biol., 30:225-234, 1971.
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
Johannes Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci. 39 (2016), no. 3, 394–404. MR 3454184, DOI 10.1002/mma.3489
C. Liu et. al, Sequential establishment of stripe patterns in an expanding cell population, Science, 334:238–241, 2011.
Toshitaka Nagai, Takasi Senba, and Kiyoshi Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), no. 3, 411–433. MR 1610709
Koichi Osaki and Atsushi Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac. 44 (2001), no. 3, 441–469. MR 1893940
Christian Stinner and Michael Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3727–3740. MR 2833007, DOI 10.1016/j.nonrwa.2011.07.006
Youshan Tao and Michael Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations 257 (2014), no. 3, 784–815. MR 3208091, DOI 10.1016/j.jde.2014.04.014
Youshan Tao and Michael Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal. 47 (2015), no. 6, 4229–4250. MR 3419885, DOI 10.1137/15M1014115
Youshan Tao and Michael Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci. 27 (2017), no. 9, 1645–1683. MR 3669835, DOI 10.1142/S0218202517500282
Jianping Wang and Mingxin Wang, Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth, J. Math. Phys. 60 (2019), no. 1, 011507, 14. MR 3899907, DOI 10.1063/1.5061738
Zhian Wang and Thomas Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos 17 (2007), no. 3, 037108, 13. MR 2356982, DOI 10.1063/1.2766864
Jun Wang, Zhian Wang, and Wen Yang, Uniqueness and convergence on equilibria of the Keller-Segel system with subcritical mass, Comm. Partial Differential Equations 44 (2019), no. 7, 545–572. MR 3949126, DOI 10.1080/03605302.2019.1581804
Michael Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010), no. 12, 2889–2905. MR 2644137, DOI 10.1016/j.jde.2010.02.008
Michael Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci. 34 (2011), no. 2, 176–190. MR 2778870, DOI 10.1002/mma.1346
Michael Winkler and Tomomi Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Anal. 170 (2018), 123–141. MR 3765558, DOI 10.1016/j.na.2018.01.002
Michael Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl. (9) 100 (2013), no. 5, 748–767 (English, with English and French summaries). MR 3115832, DOI 10.1016/j.matpur.2013.01.020
Dariusz Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal. 73 (2010), no. 2, 338–349. MR 2650817, DOI 10.1016/j.na.2010.02.047
Changwook Yoon and Yong-Jung Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math. 149 (2017), 101–123. MR 3647034, DOI 10.1007/s10440-016-0089-7