Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent

Harumi Hattori; Aesha Lagha

doi:10.3934/dcds.2021071

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2021, Volume 41, Issue 11: 5141-5164. Doi: 10.3934/dcds.2021071

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Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent

Harumi Hattori^, and
Aesha Lagha

Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA

^* Corresponding author: Harumi Hattori
^* Corresponding author: Harumi Hattori

Received: November 30, 2020

Early access: April 2021

Published: November 2021

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Abstract

We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish $ L^q $ time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find $ L^q $ time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.

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Mathematics Subject Classification: Primary: 35Q31, 35Q35; Secondary: 35Q92, 76N10.

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References

[1]	D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19. doi: 10.1080/1027366042000327098.
[2]	D. Ambrosi, A. Gamba and G. Serini, Cell directional and chemotaxis in vascular morphogenesis, Bulletin of Mathematical Biology, 66 (2004), 1851-1873. doi: 10.1016/j.bulm.2004.04.004.
[3]	I. M. Bomze, Lotka-Volterra equation and replicator dynamics: A two-dimensional classification, Biological Cybernetics, 48 (1983), 201-211. doi: 10.1007/BF00318088.
[4]	I. M. Bomze, Lotka-Volterra equation and replicator dynamics: New issues in classification, Biological Cybernetics, 72 (1995), 447-453. doi: 10.1007/BF00201420.
[5]	M. Chae, K. Kang and J. Lee, Existence of smooth solutions to chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297. doi: 10.3934/dcds.2013.33.2271.
[6]	R. J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differential Equations, 8 (2011), 375-413. doi: 10.1142/S0219891611002421.
[7]	R. Duan, Q. Liu and C. Zhu, The Cauchy problem on the compressible two-fluids Euler-Maxwell Equations, SIAM J. Math. Anal., 44 (2012), 102-133. doi: 10.1137/110838406.
[8]	R. Duan, A. Lorz and P. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Diff. Equations, 35 (2010), 1635-1673. doi: 10.1080/03605302.2010.497199.
[9]	A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and burgers dynamics in blood vessels formation, Physical Review Letters, 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101.
[10]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 81-205. doi: 10.1007/BF00280740.
[11]	E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[12]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[13]	E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal. Real World Appl., 46 (2019), 421-445. doi: 10.1016/j.nonrwa.2018.09.012.
[14]	J. Liu and Z.-A. Wang, Classical solutions and steady states of attraction-repulsion chemotaxis in one dimension, Journal of Biological Dynamics, 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722.
[15]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.
[16]	A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.
[17]	A. D. Rodriguez, L. C. F. Ferreira and É. J. Villamizar-Roa, Global existence for an attraction-repulsion chemotaxis fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 423-447. doi: 10.3934/dcdsb.2018180.
[18]	G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, EMBO Journal, 22 (2003), 1771-1779. doi: 10.1093/emboj/cdg176.
[19]	Z. Tan and J. Zhou, Global existence and time decay estimate of solutions to the Keller- Segel system, Math. Meth. Appl. Sci., 42 (2019), 375-402. doi: 10.1002/mma.5352.
[20]	Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Journal of Differential Equations, 2016 (2016), 21 pp.