Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility

Xu Song; Jingyu Li; Xu Song; Jingyu Li

doi:10.3934/mbe.2022652

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 13988-14028. doi: 10.3934/mbe.2022652

Previous Article Next Article

Research article Special Issues

Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility

Xu Song ,
Jingyu Li ^,

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Academic Editor: Zhi-An Wang

Received: 01 August 2022 Revised: 05 September 2022 Accepted: 07 September 2022 Published: 22 September 2022

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.
- chemotaxis,
- singular sensitivity,
- signal-suppressed motility,
- spike,
- asymptotic stability
Citation: Xu Song, Jingyu Li. Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13988-14028. doi: 10.3934/mbe.2022652

Related Papers:

Abstract

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.

References

[1]	S. Choi, Y. Kim, Chemotactic traveling waves by metric of food, SIAM J. Appl. Math., 75 (2015), 2268–2289. https://doi.org/10.1137/15100429X doi: 10.1137/15100429X
[2]	S. Choi, Y. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090. https://doi.org/10.1016/j.jmaa.2020.124090 doi: 10.1016/j.jmaa.2020.124090
[3]	J. Adler, Chemotaxis in bacteria: Motile Escherichia coli migrate in bands that are influenced by oxygen and organic nutrients, Science, 153 (1966), 708–716. http://dx.doi.org/10.1126/science.153.3737.708 doi: 10.1126/science.153.3737.708
[4]	E. F. Keller, L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. https://doi.org/10.1016/0022-5193(71)90051-8 doi: 10.1016/0022-5193(71)90051-8
[5]	J. Ahn, S. Choi, M. Yoo, Global wellposedness of nutrient-taxis systems derived by a food metric, Discrete Contin. Dyn. Syst., 41 (2021), 6001–6022. https://doi.org/10.3934/dcds.2021104 doi: 10.3934/dcds.2021104
[6]	J. Ahn, S. Choi, M. Yoo, Global classical solvability in a food metric chemotaxis system under zero boundary conditions at infinity, Nonlinear Anal., 224 (2022), 113083. https://doi.org/10.1016/j.na.2022.113083 doi: 10.1016/j.na.2022.113083
[7]	I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, in Proceedings of the National Academy of Sciences of the United States of America, PNAS, 102 (2005), 2277–2282. https://doi.org/10.1073/pnas.0406724102
[8]	J. Ahn, C. Yoon, Global well-posedness and stability of constant equilibria in parabolic elliptic chemotaxis systems without gradient sensing, Nonlinearity, 32 (2019), 1327–1351. https://doi.org/10.1088/1361-6544/aaf513 doi: 10.1088/1361-6544/aaf513
[9]	C. Yoon, Y. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101–123. https://doi.org/10.1007/s10440-016-0089-7 doi: 10.1007/s10440-016-0089-7
[10]	K. Fujie, J. Jiang, Global existence for a kinetic model of pattern formation with density-suppressed motilities, J. Differ. Equations, 269 (2020), 5338–5378. https://doi.org/10.1016/j.jde.2020.04.001 doi: 10.1016/j.jde.2020.04.001
[11]	H. Jin, Z. A. Wang, Crtical mass on the Keller-Segel system with signal-dependent motility, in Proceedings of the American Mathematical Society, AMS, 148 (2020), 4855–4873. https://doi.org/10.1090/proc/15124
[12]	Z. A. Wang, X. Xu, Steady states and pattern formation of the density-suppressed motility model, IMA J. Appl. Math., 86 (2021), 577–603. https://doi.org/10.1093/imamat/hxab006 doi: 10.1093/imamat/hxab006
[13]	J. A. Carrillo, J. Li, Z. A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability, in Proceedings of the London Mathematical Society, LMS, 122 (2021), 42–68. https://doi.org/10.1112/plms.12319
[14]	C. S. Lin, W. M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1988), 1–27. https://doi.org/10.1016/0022-0396(88)90147-7 doi: 10.1016/0022-0396(88)90147-7
[15]	W. M. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819–851. https://doi.org/10.1002/cpa.3160440705 doi: 10.1002/cpa.3160440705
[16]	Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521–529. https://doi.org/10.1016/j.jmaa.2011.02.041 doi: 10.1016/j.jmaa.2011.02.041
[17]	K. Baghaei, A. Khelghati, Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant, C. R. Math., 355 (2017), 633–639. https://doi.org/10.1016/j.crma.2017.04.009 doi: 10.1016/j.crma.2017.04.009
[18]	S. Frassu, G. Viglialoro, Boundedness criteria for a class of indirect (and direct) chemotaxis-consumption models in high dimensions, Appl. Math. Lett., 132 (2022), 108108. https://doi.org/10.1016/j.aml.2022.108108 doi: 10.1016/j.aml.2022.108108
[19]	D. Li, J. Zhao, Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility, Z. Angew. Math. Phys., 72 (2021), 57. https://doi.org/10.1007/s00033-021-01493-y doi: 10.1007/s00033-021-01493-y
[20]	L. Wang, Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility, Appl. Math. Lett., 125 (2022), 107724. https://doi.org/10.1016/j.aml.2021.107724 doi: 10.1016/j.aml.2021.107724
[21]	M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987–1024. https://doi.org/10.1142/S0218202516500238 doi: 10.1142/S0218202516500238
[22]	M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differ. Equations, 264 (2018), 2310–2350. https://doi.org/10.1016/j.jde.2017.10.029 doi: 10.1016/j.jde.2017.10.029
[23]	G. Hong, Z. A. Wang, Asymptotic stability of exogenous chemotaxis systems with physical boundary conditions, Quart. Appl. Math., 79 (2021), 717–743. https://doi.org/10.1090/qam/1599 doi: 10.1090/qam/1599
[24]	M. Braukhoff, J. Lankeit, Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen, Math. Models Methods Appl. Sci., 29 (2019), 2033–2062. https://doi.org/10.1142/S0218202519500398 doi: 10.1142/S0218202519500398
[25]	C. C. Lee, Z. A. Wang, W. Yang, Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis, Nonlinearity, 33 (2020), 5111–5141. https://doi.org/10.1088/1361-6544/ab8f7c doi: 10.1088/1361-6544/ab8f7c
[26]	M. Fuest, J. Lankeit, M. Mizukami, Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model, J. Differ. Equations, 271 (2021), 254–279. https://doi.org/10.1016/j.jde.2020.08.021 doi: 10.1016/j.jde.2020.08.021
[27]	J. Li, T. Li, Z. A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819–2849. https://doi.org/10.1142/s0218202514500389 doi: 10.1142/s0218202514500389
[28]	H. A. Levine, B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683–730. https://epubs.siam.org/doi/abs/10.1137/S0036139995291106 doi: 10.1137/S0036139995291106
[29]	J. Li, Z. A. Wang, Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space, J. Differ. Equations, 268 (2020), 6940–6970. https://doi.org/10.1016/j.jde.2019.11.076 doi: 10.1016/j.jde.2019.11.076
[30]	T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dnamics, Publ. Math. D'Orsay, 79 (1978), 46–53.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)