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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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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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##### 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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