Abstract
This paper is concerned with the density-suppressed motility model: \(u_{t}=\Delta \left( \displaystyle \frac{u^m}{v^\alpha }\right) +\beta uf(w), v_{t}=D\Delta v-v+u, w_{t}=\Delta w-uf(w)\) in a smoothly bounded convex domain \(\Omega \subset {{\mathbb {R}}}^2\), where \(m>1\), \(\alpha>0, \beta >0\) and \(D>0\) are parameters, the response function f satisfies \(f\in C^1([0,\infty )), f(0)=0, f(w)>0\) in \((0,\infty )\). This system describes the density-suppressed motility of Eeshcrichia coli cells in the process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large D the problem admits at least one global weak solution (u, v, w) which will asymptotically converge to the spatially uniform equilibrium \((\overline{u_0}+\beta \overline{w_0},\overline{u_0}+\beta \overline{w_0},0)\) with \(\overline{u_0}=\frac{1}{|\Omega |}\int _{\Omega }u(x,0)dx \) and \(\overline{w_0}=\frac{1}{|\Omega |}\int _{\Omega }w(x,0)dx \) in \(L^\infty (\Omega )\).
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References
Ahn, J., Yoon, C.: Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis system with gradient sensing. Nonlinearity 32, 1327–1351 (2019)
Bellomo, N., Bellouquid, A., Chouhad, N.: From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid. Math. Models Methods Appl. Sci. 26, 2041–2069 (2016)
Calvez, V., Carrillo, J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86(9), 155–175 (2006)
Cañizo, J.A., Desvillettes, L., Fellner, K.: Improved duality estimates and applications to reaction-diffusion equations. Commun. PDE. 39, 1185–1284 (2014)
Fu, X., Tang, L., Liu, C., Huang, J., Hwa, T., Lenz, P.: Stripe formation in bacterial system with density-suppressed motility. Phys. Rev. Lett. 108, 198102 (2012)
Fujie, K.: Study of reaction–diffusion systems modeling chemotaxis, Doctoral thesis (2016)
Fujie, K., Jiang, J.: Global existence for a kinetic model of pattern formation with density-suppressed motilities. J. Differ. Equ. 269, 5338–5378 (2020)
Fujie, K., Jiang, J.: Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities. Calc. Var. Partial Differ. Equ. 60, 1–37 (2021)
Herrero, M.A., Velázquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623 (1996)
Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scu. Norm. Super. Pisa Cl. Sci. 24, 663–683 (1997)
Hillen, T., Painter, K.J., Winkler, M.: Convergence of a cancer invasion model to a logistic chemotaxis model. Math. Models Methods Appl. Sci. 23, 165–198 (2013)
Ishida, S., Seki, K., Yokota, T.: Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993–3010 (2014)
Isenbach, M.: Chemotaxis. Imperial College Pres, London (2004)
Jiang, J., Laurenot, P.: Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility, preprint arXiv:2101.10666
Jin, H., Kim, Y.J., Wang, Z.: Boundedness, stabilization and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math. 78(3), 1632–1657 (2018)
Jin, H., Shi, S., Wang, Z.: Boundedness and asymptotics of a reaction-diffusion system with density-dependent motility. J. Differ. Equ. 269, 6758–6793 (2020)
Keller, E., Segel, L.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Qquasi-linear Equations of Parabolic Type, Transl. Math. Monogr., vol. 23. American Mathematical Society, Providence (1968)
Liu, C., et al.: Sequential establishment of stripe patterns in an expanding cell population. Science 334, 238–241 (2011)
Lv, W.B., Wang, Q.: Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source. Z. Angew. Math. Phys. 71(2), 53 (2020)
Lv, W.B., Wang, Q.: A n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization. Proc. R. Soc. Edinb. A (2020). https://doi.org/10.1017/prm.2020.38
Lv, W.B., Wang, Z.A.: Global classical solutions for a class of reaction-diffusion system with density-suppressed motility. arXiv:2102.08042
Ma, M., Peng, R., Wang, Z.: Stationary and non-stationary patterns of the density-suppressed motility model. Phys. D 402, 132559 (2020)
Murray, J.D.: Mathematical Biology. Springer, New York (2001)
Porzio, M.M., Vespri, V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103(1), 146–178 (1993)
Rothe, F.: Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol. 1072. Springer, Berlin (1984)
Stinner, C., Surulescu, C., Winkler, M.: Global weak solutions in a PDE-ODE system modelling multiscale cancer cell invasion. SIAM J. Math. Anal. 46(3), 1969–2007 (2014)
Tao, Y., Winkler, M.: Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system. Math. Model Meth. Appl. Sci. 27, 1645–1683 (2017)
Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)
Vázquez, J.L.: The Porous Medium Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (2007)
Wang, J., Wang, M.: Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth. J. Math. Phys. 60, 011507 (2019)
Winkler, M.: Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity. Nonlinearity 30, 735–764 (2017)
Winkler, M.: Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities. J. Differ. Equ. 266, 8034–8066 (2019)
Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J. Differ. Equ. 248(12), 2889–2905 (2010)
Winkler, M.: Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Calc. Var. Partial Diff. Equ. 54, 3789–3828 (2015)
Winkler, M.: Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller-Segel type systems involving signal-dependent motilities? Nonlinearity 33(12), 6590–6632 (2020)
Yoon, C., Kim, Y.J.: Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion. Acta Appl. Math. 149, 101–123 (2017)
Acknowledgements
The authors would like to express their gratitude to the anonymous referee for the careful reading with useful comments to improve the manuscript. The authors thank Professor Michael Winkler for his helpful comments. This work is supported by the NNSF of China (No.12071030) and Beijing key laboratory on MCAACI.
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Communicated by P. H. Rabinowitz.
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Xu, C., Wang, Y. Asymptotic behavior of a quasilinear Keller–Segel system with signal-suppressed motility. Calc. Var. 60, 183 (2021). https://doi.org/10.1007/s00526-021-02053-y
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DOI: https://doi.org/10.1007/s00526-021-02053-y