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

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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.

Keywords: Large time behavior, chemotaxis-growth system, nonlinear cue production.

Mathematics Subject Classification: Primary: 35B40, 35K57, 35Q92, 92C1.

Citation: Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353. doi: 10.1007/BF00249679. Google Scholar
[6]	M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. Google Scholar
[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. Google Scholar
[8]	M. R. Myerscough, P. K. Maini and J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353. doi: 10.1007/BF00249679. Google Scholar
[6]	M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. Google Scholar
[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. Google Scholar
[8]	M. R. Myerscough, P. K. Maini and J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.Google Scholar
[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. Google Scholar
[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. Google Scholar

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