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September  2016, 36(9): 5025-5046. doi: 10.3934/dcds.2016018

Global dynamics in a fully parabolic chemotaxis system with logistic source

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received  July 2015 Revised  January 2016 Published  May 2016

In this paper, we consider a fully parabolic chemotaxis system \begin{eqnarray*}\label{1} \left\{ \begin{array}{llll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+u-\mu u^r,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain $\Omega\subset R^n(n=2,3)$, where $\chi>0, \mu>0$ and $r\geq 2$.
    For the dimensions $n=2$ and $n=3$, we establish results on the global existence and boundedness of classical solutions to the corresponding initial-boundary problem, provided that $\chi$, $\mu$ and $r$ satisfy some explicit conditions. Apart from this, we also show that if $\frac{\mu^{\frac{1}{r-1}}}{\chi}>20$ and $r\geq 2$ and $r\in \mathbb{N}$ the solution of the system approaches the steady state $\left(\mu^{-\frac{1}{r-1}}, \mu^{-\frac{1}{r-1}}\right)$ as time tends to infinity.
Citation: Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
References:
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N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues,, Math. Models Methods Appl. Sci., 25 (2015), 1663.  doi: 10.1142/S021820251550044X.  Google Scholar

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P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III,, Colloq. Mathematicum, 68 (1995), 229.   Google Scholar

[4]

X. R. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces,, Discrete Cont. Dyns. S-A., 35 (2015), 1891.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

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X. R. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model,, Zeitschrift für angewandte Mathematik und Physik, 67 (2016).  doi: 10.1007/s00033-015-0601-3.  Google Scholar

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T. Cieślak and P. H. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré Anal. Non Linéaire, 27 (2010), 437.  doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

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T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[8]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models,, J. Differential Equations, 258 (2015), 2080.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

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T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

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A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks,, J. Math. Anal. Appl., 272 (2002), 138.  doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

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H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

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M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633.   Google Scholar

[13]

D. Horstemann, 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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[15]

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.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[16]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Am. Math. Soc., 329 (1992), 819.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[17]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[19]

Y. H. Li, K. Lin and C. L. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxis-haptotaxis model in high dimensions,, Appl. Math. Lett., 50 (2015), 91.  doi: 10.1016/j.aml.2015.06.010.  Google Scholar

[20]

J. Lankeit, Chemotaxis can prevent thresholds on population density,, Discrete Cont. Dyns. S-B., 20 (2015), 1499.  doi: 10.3934/dcdsb.2015.20.1499.  Google Scholar

[21]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source,, J. Differential Equations, 258 (2015), 1158.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[22]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,, J. Differential Equations, 103 (1993), 146.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[23]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[24]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar

[25]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411.   Google Scholar

[26]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion,, SIAM J. MAth. Anal., 46 (2014), 3761.  doi: 10.1137/140971853.  Google Scholar

[27]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant,, J. Differential Equations, 258 (2015), 1592.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar

[28]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[29]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441.   Google Scholar

[30]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , ().   Google Scholar

[31]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[32]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[33]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model,, Nonlinearity, 27 (2014), 1225.  doi: 10.1088/0951-7715/27/6/1225.  Google Scholar

[34]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant,, J. Differential Equations, 257 (2014), 784.  doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[35]

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.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[36]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source,, J. Differential Equations, 259 (2015), 6142.  doi: 10.1016/j.jde.2015.07.019.  Google Scholar

[37]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849.  doi: 10.1080/03605300701319003.  Google Scholar

[38]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties,, J. Math. Anal Appl, 348 (2008), 708.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516.  doi: 10.1080/03605300903473426.  Google Scholar

[40]

M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[41]

M. Winkler, Does a volume-filling effect always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

[42]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[43]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?,, J. Nonlinear Sci., 24 (2014), 809.  doi: 10.1007/s00332-014-9205-x.  Google Scholar

[44]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening,, J. Differential Equations, 257 (2014), 1056.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[45]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal., 211 (2014), 455.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[46]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity,, Calc. Var. Partial Differential Equations, 54 (2015), 3789.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[47]

M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues,, Math. Models Methods Appl. Sci., 25 (2015), 1663.  doi: 10.1142/S021820251550044X.  Google Scholar

[3]

P. Biler, Existence and nonexistence of solutions for a model of gravitational interaction of particles III,, Colloq. Mathematicum, 68 (1995), 229.   Google Scholar

[4]

X. R. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces,, Discrete Cont. Dyns. S-A., 35 (2015), 1891.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[5]

X. R. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model,, Zeitschrift für angewandte Mathematik und Physik, 67 (2016).  doi: 10.1007/s00033-015-0601-3.  Google Scholar

[6]

T. Cieślak and P. H. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré Anal. Non Linéaire, 27 (2010), 437.  doi: 10.1016/j.anihpc.2009.11.016.  Google Scholar

[7]

T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.  doi: 10.1016/j.jde.2012.01.045.  Google Scholar

[8]

T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models,, J. Differential Equations, 258 (2015), 2080.  doi: 10.1016/j.jde.2014.12.004.  Google Scholar

[9]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[10]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks,, J. Math. Anal. Appl., 272 (2002), 138.  doi: 10.1016/S0022-247X(02)00147-6.  Google Scholar

[11]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[12]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633.   Google Scholar

[13]

D. Horstemann, 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.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[14]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[15]

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.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[16]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Am. Math. Soc., 329 (1992), 819.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[17]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[18]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379.  doi: 10.1016/j.jmaa.2008.01.005.  Google Scholar

[19]

Y. H. Li, K. Lin and C. L. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxis-haptotaxis model in high dimensions,, Appl. Math. Lett., 50 (2015), 91.  doi: 10.1016/j.aml.2015.06.010.  Google Scholar

[20]

J. Lankeit, Chemotaxis can prevent thresholds on population density,, Discrete Cont. Dyns. S-B., 20 (2015), 1499.  doi: 10.3934/dcdsb.2015.20.1499.  Google Scholar

[21]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source,, J. Differential Equations, 258 (2015), 1158.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[22]

M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations,, J. Differential Equations, 103 (1993), 146.  doi: 10.1006/jdeq.1993.1045.  Google Scholar

[23]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[24]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar

[25]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj. Ser. Int., 40 (1997), 411.   Google Scholar

[26]

M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion,, SIAM J. MAth. Anal., 46 (2014), 3761.  doi: 10.1137/140971853.  Google Scholar

[27]

M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant,, J. Differential Equations, 258 (2015), 1592.  doi: 10.1016/j.jde.2014.11.009.  Google Scholar

[28]

K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[29]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkc. Ekvacioj. Ser. Int., 44 (2001), 441.   Google Scholar

[30]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , ().   Google Scholar

[31]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[32]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520.  doi: 10.1016/j.jde.2011.07.010.  Google Scholar

[33]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model,, Nonlinearity, 27 (2014), 1225.  doi: 10.1088/0951-7715/27/6/1225.  Google Scholar

[34]

Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant,, J. Differential Equations, 257 (2014), 784.  doi: 10.1016/j.jde.2014.04.014.  Google Scholar

[35]

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.  doi: 10.1007/s00033-015-0541-y.  Google Scholar

[36]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source,, J. Differential Equations, 259 (2015), 6142.  doi: 10.1016/j.jde.2015.07.019.  Google Scholar

[37]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849.  doi: 10.1080/03605300701319003.  Google Scholar

[38]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and boundedness properties,, J. Math. Anal Appl, 348 (2008), 708.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516.  doi: 10.1080/03605300903473426.  Google Scholar

[40]

M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[41]

M. Winkler, Does a volume-filling effect always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.  doi: 10.1002/mma.1146.  Google Scholar

[42]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[43]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?,, J. Nonlinear Sci., 24 (2014), 809.  doi: 10.1007/s00332-014-9205-x.  Google Scholar

[44]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening,, J. Differential Equations, 257 (2014), 1056.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[45]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system,, Arch. Ration. Mech. Anal., 211 (2014), 455.  doi: 10.1007/s00205-013-0678-9.  Google Scholar

[46]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity,, Calc. Var. Partial Differential Equations, 54 (2015), 3789.  doi: 10.1007/s00526-015-0922-2.  Google Scholar

[47]

M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044.  doi: 10.1016/j.na.2009.07.045.  Google Scholar

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