# American Institute of Mathematical Sciences

March  2017, 22(2): 465-475. doi: 10.3934/dcdsb.2017021

## Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals

 1 College of Information Science & Technology, Dong Hua University, Shanghai 200051, China 2 Department of Applied Mathematics, Dong Hua University, Shanghai 200051, China

Received  March 2016 Revised  April 2016 Published  December 2016

We consider the chemotaxis-growth system
 $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi \nabla \cdot (u\nabla v) + \mu u(1 - u),}&{x \in \Omega ,{\mkern 1mu} t > 0,}\\ {{v_t} = \Delta v - v + h(u),}&{x \in \Omega ,{\mkern 1mu} t > 0,} \end{array}} \right.$
under no-flux boundary conditions, in a convex bounded domain
 $Ω\subset\mathbb{R}^3$
with smooth boundary, where
 $χ>0$
and
 $μ>0$
are given parameters, and
 $h(s)$
is a prescribed function on
 $[0, ∞)$
.
It is shown that under the assumption that
 $4|{h}'|<\sqrt{2\mu -7{{\chi }^{2}}},$
for any given nonnegative
 $u_0∈ C^0(\bar{Ω})$
and
 $v_0∈ W^{1, ∞}(Ω)$
the system possesses a global classical solution which is bounded in
 $Ω× (0, ∞)$
. Moreover, whenever
 $χ |h'| < \sqrt{8μ},$
any bounded classical solution constructed above stabilizes to the constant stationary solution
 $(1, h(1))$
as the time goes to infinity.
Citation: Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021
##### References:
 [1] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.  doi: 10.1016/j.aml.2015.12.001. [2] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [3] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028. [4] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [5] P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353.  doi: 10.1007/BF00249679. [6] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. [7] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007. [8] M. R. Myerscough, P. K. Maini and J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. [9] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. B, 18 (2013), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627. [10] E. Nakaguchi and K. Osaki, $L_p$-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.  doi: 10.1619/fesi.59.51. [11] E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint. [12] M. E. Orme and M. A. J. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol., 13 (1996), 73-98. [13] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [14] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [15] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [16] Y. Tao and M. Winkler, Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [17] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y. [18] Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint. [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

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##### References:
 [1] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.  doi: 10.1016/j.aml.2015.12.001. [2] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [3] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028. [4] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [5] P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353.  doi: 10.1007/BF00249679. [6] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. [7] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007. [8] M. R. Myerscough, P. K. Maini and J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. [9] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. B, 18 (2013), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627. [10] E. Nakaguchi and K. Osaki, $L_p$-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.  doi: 10.1619/fesi.59.51. [11] E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint. [12] M. E. Orme and M. A. J. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol., 13 (1996), 73-98. [13] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [14] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045. [15] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [16] Y. Tao and M. Winkler, Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [17] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y. [18] Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint. [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
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