### AIMS Mathematics

2016, Issue 3: 156-164. doi: 10.3934/Math.2016.3.156
Research article

# Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system

• Received: 21 July 2016 Accepted: 03 August 2016 Published: 10 August 2016
• This paper deals with the two-species chemotaxis system $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{v_t} = \Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - v)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{w_t} = d\Delta w + h(u,v,w)}&{{\text{in}}\;\Omega \times (0,\infty ),} \end{array}} \right.$ where $\Omega$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, $N\in \mathbb{N}$; $h$, $\chi_i$ are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.

Citation: Masaaki Mizukami. Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system[J]. AIMS Mathematics, 2016, 1(3): 156-164. doi: 10.3934/Math.2016.3.156

### Related Papers:

• This paper deals with the two-species chemotaxis system $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1 - u)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{v_t} = \Delta v - \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1 - v)}&{{\text{in}}\;\Omega \times (0,\infty ),} \\ {{w_t} = d\Delta w + h(u,v,w)}&{{\text{in}}\;\Omega \times (0,\infty ),} \end{array}} \right.$ where $\Omega$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, $N\in \mathbb{N}$; $h$, $\chi_i$ are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.
 [1] X. Bai, M.Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65 (2016), 553-583. [2] 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), 1663-1763. [3] C. Bianca, M. Pennisi, S. Motta, M.A. Ragusa, Immune system network and cancer vaccine. AIP Conf. Proc., 1389 (2011), 945-948. [4] T. Hillen, K. J. Painter, A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009), 183-217. [5] D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. Jahresber. Deutsch. Math. -Verein. 106 (2004), 51-69. [6] D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species. J. Nonlinear Sci. 21 (2011), 231-270. [7] S. Kathirvel, R. Jangre, S. Ko, Design of a novel energy eficient topology for maximum magnitude generator. IET Computers and Digital Techniques, 10 (2016), 93101. [8] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. [9] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, 1968. [10] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. submitted. [11] M. Mizukami, T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diflusion. J. Diflerential Equations 261 (2016), 2650-2669. [12] M. Negreanu, J. I. Tello, On a two species chemotaxis model with slow chemical diflusion. SIAM J. Math. Anal. 46 (2014), 3761-3781. [13] M. Negreanu, J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diflusive chemoattractant. J. Diflerential Equations 258 (2015), 1592-1617. [14] F. Pappalardo, V. Brusic, F. Castiglione, C. Schonbach, Computational and bioinforfatics techniques for immunology. BioMed research international, 2014 (2014), 1-2. [15] G. Wolansky, Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13 (2002), 641-661.

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