# American Institute of Mathematical Sciences

November  2021, 41(11): 5141-5164. doi: 10.3934/dcds.2021071

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

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

* Corresponding author: Harumi Hattori

Received  December 2020 Published  November 2021 Early access  April 2021

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.

Citation: Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071
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##### References:
 [1] Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322 [2] Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032 [3] Guochun Wu, Han Wang, Yinghui Zhang. Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in $\mathbb R^3$. Electronic Research Archive, , () : -. doi: 10.3934/era.2021067 [4] Jan-Cornelius Molnar. On two-sided estimates for the nonlinear Fourier transform of KdV. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3339-3356. doi: 10.3934/dcds.2016.36.3339 [5] Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007 [6] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [7] Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021041 [8] Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407 [9] Ryo Ikehata, Shingo Kitazaki. Optimal energy decay rates for some wave equations with double damping terms. Evolution Equations & Control Theory, 2019, 8 (4) : 825-846. doi: 10.3934/eect.2019040 [10] Ruy Coimbra Charão, Jáuber Cavalcante Oliveira, Gustavo Alberto Perla Menzala. Energy decay rates of magnetoelastic waves in a bounded conductive medium. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 797-821. doi: 10.3934/dcds.2009.25.797 [11] Petronela Radu, Grozdena Todorova, Borislav Yordanov. Higher order energy decay rates for damped wave equations with variable coefficients. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 609-629. doi: 10.3934/dcdss.2009.2.609 [12] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 [13] Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002 [14] Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258 [15] Linghai Zhang. Decay estimates with sharp rates of global solutions of nonlinear systems of fluid dynamics equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2181-2200. doi: 10.3934/dcdss.2016091 [16] Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75 [17] Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005 [18] Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579 [19] Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69 [20] Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419

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