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Asymptotic behaviors of GreenSch potentials at infinity and its applications
Boundedness and asymptotic stability in a twospecies chemotaxiscompetition model with signaldependent sensitivity
Department of Mathematics, Tokyo University of Science, 13 Kagurazaka, Shinjukuku, Tokyo 1628601, Japan 
$\left\{ {\begin{array}{*{20}{l}}{{u_t} = {d_1}\Delta u  \nabla \cdot (u{\chi _1}(w)\nabla w) + {\mu _1}u(1  u  {a_1}v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{v_t} = {d_2}\Delta v  \nabla \cdot (v{\chi _2}(w)\nabla w) + {\mu _2}v(1  {a_2}u  v)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\\{{w_t} = {d_3}\Delta w + h(u,v,w)}&{;{\rm{in}}\;\Omega \times (0,\infty ),}\end{array}} \right.$ 
$\Omega$ 
$\mathbb{R}^n$ 
$\partial \Omega$ 
$n\in \mathbb{N}$ 
$h$ 
$\chi_i$ 
$\chi_i(w)=\chi_i$ 
$\mu_1, \mu_2$ 
$\mu_1, \mu_2$ 
References:
[1] 
X. Bai, M. Winkler, Equilibration in a fully parabolic twospecies chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553583. 
[2] 
N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 16631763. 
[3] 
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. SpringerVerlag, BerlinNew York, 1981. 
[4] 
T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183217. 
[5] 
D. Horstmann, From 1970 until present: the KellerSegel model in chemotaxis and its consequences, Jahresber. Deutsch. Math. Verein., 106 (2004), 5169. 
[6] 
D. Horstmann, M. Winkler, Boundedness vs. blowup in a chemotaxis system, J. Differential Equations, 215 (2005), 52107. 
[7] 
D. Horstmann, Generalizing the KellerSegel model: Lyapunov functionals, steady state analysis, and blowup results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231270. 
[8] 
E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399415. 
[9] 
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 1968. 
[10] 
M. Mizukami, T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 26502669. 
[11] 
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signaldependent sensitivity, Math. Nachr., 283 (2010), 16641673. 
[12] 
M. Winkler, Boundedness in the higherdimensional parabolicparabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 15161537. 
[13] 
G. Wolansky, Multicomponents chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641661. 
[14] 
Q. Zhang, Y. Li, Global boundedness of solutions to a twospecies chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 8393. 
show all references
References:
[1] 
X. Bai, M. Winkler, Equilibration in a fully parabolic twospecies chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553583. 
[2] 
N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 16631763. 
[3] 
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. SpringerVerlag, BerlinNew York, 1981. 
[4] 
T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183217. 
[5] 
D. Horstmann, From 1970 until present: the KellerSegel model in chemotaxis and its consequences, Jahresber. Deutsch. Math. Verein., 106 (2004), 5169. 
[6] 
D. Horstmann, M. Winkler, Boundedness vs. blowup in a chemotaxis system, J. Differential Equations, 215 (2005), 52107. 
[7] 
D. Horstmann, Generalizing the KellerSegel model: Lyapunov functionals, steady state analysis, and blowup results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231270. 
[8] 
E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399415. 
[9] 
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, 1968. 
[10] 
M. Mizukami, T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 26502669. 
[11] 
M. Winkler, Absence of collapse in a parabolic chemotaxis system with signaldependent sensitivity, Math. Nachr., 283 (2010), 16641673. 
[12] 
M. Winkler, Boundedness in the higherdimensional parabolicparabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 15161537. 
[13] 
G. Wolansky, Multicomponents chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641661. 
[14] 
Q. Zhang, Y. Li, Global boundedness of solutions to a twospecies chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 8393. 
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