Global dynamics in a fully parabolic chemotaxis system with logistic source

Ke  Lin; Chunlai Mu

doi:10.3934/dcds.2016018

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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

Ke Lin ^1, and Chunlai Mu ^1,

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

Received July 2015 Revised January 2016 Published May 2016

Abstract
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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.

Keywords: logistic source, Chemotaxis, boundedness, asymptotic stability..

Mathematics Subject Classification: Primary: 35K35, 92C17; Secondary: 35K55, 35B3.

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:

[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

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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American Institute of Mathematical Sciences

Global dynamics in a fully parabolic chemotaxis system with logistic source

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