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June  2022, 42(6): 2893-2925. doi: 10.3934/dcds.2022003

## Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$

 1 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

*Corresponding author: Shuwen Xue (szx0006@auburn.edu; sxue@mun.ca)

Received  March 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on
 $\mathbb{R}^{N}$
,
 $$$\begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)$$$
where
 $\chi, \ a,\ b,\ \lambda,\ \mu$
are positive constants and
 $N$
is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption
 $b>\frac{N\chi\mu}{4}$
, the global existence of a unique classical solution
 $(u(x,t;u_0, v_0),v(x,t;u_0, v_0))$
of (1) with
 $u(x,0;u_0, v_0) = u_0(x)$
and
 $v(x,0;u_0, v_0) = v_0(x)$
for every nonnegative, bounded, and uniformly continuous function
 $u_0(x)$
, and every nonnegative, bounded, uniformly continuous, and differentiable function
 $v_0(x)$
. Next, under the same assumption
 $b>\frac{N\chi\mu}{4}$
, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function
 $u_0$
is bounded below by a positive constant independent of
 $(u_0, v_0)$
when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function
 $u_0$
. We show that there is
 $K = K(a,\lambda,N)>\frac{N}{4}$
such that if
 $b>K \chi\mu$
and
 $\lambda\geq \frac{a}{2}$
, then for every strictly positive initial function
 $u_0(\cdot)$
, it holds that
 $\lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0.$
Citation: Wenxian Shen, Shuwen Xue. Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^{N}$. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2893-2925. doi: 10.3934/dcds.2022003
##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [2] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9. [3] J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Adv. Math. Sci. Appl., 5 (1995), 659-680. [4] J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on ${{\mathbb R}}^{N}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389. [5] E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [7] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. [9] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [10] T. B. Issa and W. Shen, Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments, J. Math. Anal. Appl., 490 (2020), 124204, 30 pp. doi: 10.1016/j.jmaa.2020.124204. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [13] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discr. Cont. Dyn. Syst. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [14] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [15] D. Li, C. Mu, K. Lin and L. Wang, Large time behavior of solutions to an attraction-repulsion chemotaxis system with logistic source in three demensions, J. Math. Anal. Appl., 448 (2017), 914-936.  doi: 10.1016/j.jmaa.2016.11.036. [16] J. Li, Y. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002. [17] K. Lin and C. L. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [18] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser Inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [19] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in ${{\mathbb R}}^N$., Funkcial. Ekvac., 46 (2003), 383-407.  doi: 10.1619/fesi.46.383. [20] T. Nagai and T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl., 336 (2007), 704-726.  doi: 10.1016/j.jmaa.2007.03.014. [21] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [22] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [23] K. J. Painter, Mathematical models for chemotaxis1 and their applications in self organization phenomena, J. Theoret. Biol., 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019. [24] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011. [27] R. B. Salako and W. Shen, Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268. [28] R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time-dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146. [29] R. B. Salako, W. Shen and S. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490.  doi: 10.1007/s00285-019-01400-0. [30] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Eq., 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019. [31] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [32] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [33] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [34] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Part. Differential Eq., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [35] 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. [36] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057. [37] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [38] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Eq., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [39] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [40] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 1125–1133. doi: 10.3934/proc.2015.1125. [41] J. Zheng, Y. Y. Li, G. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064. [42] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031.

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##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [2] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9. [3] J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Adv. Math. Sci. Appl., 5 (1995), 659-680. [4] J. I. Diaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on ${{\mathbb R}}^{N}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389. [5] E. Galakhov, O. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008. [6] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981. [7] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3. [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. [9] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [10] T. B. Issa and W. Shen, Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments, J. Math. Anal. Appl., 490 (2020), 124204, 30 pp. doi: 10.1016/j.jmaa.2020.124204. [11] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [12] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6. [13] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discr. Cont. Dyn. Syst. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [14] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [15] D. Li, C. Mu, K. Lin and L. Wang, Large time behavior of solutions to an attraction-repulsion chemotaxis system with logistic source in three demensions, J. Math. Anal. Appl., 448 (2017), 914-936.  doi: 10.1016/j.jmaa.2016.11.036. [16] J. Li, Y. Ke and Y. Wang, Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.  doi: 10.1016/j.nonrwa.2017.07.002. [17] K. Lin and C. L. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018. [18] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser Inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [19] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in ${{\mathbb R}}^N$., Funkcial. Ekvac., 46 (2003), 383-407.  doi: 10.1619/fesi.46.383. [20] T. Nagai and T. Yamada, Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space, J. Math. Anal. Appl., 336 (2007), 704-726.  doi: 10.1016/j.jmaa.2007.03.014. [21] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [22] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [23] K. J. Painter, Mathematical models for chemotaxis1 and their applications in self organization phenomena, J. Theoret. Biol., 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019. [24] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Phys. D, 240 (2011), 363-375.  doi: 10.1016/j.physd.2010.09.011. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [26] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011. [27] R. B. Salako and W. Shen, Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268. [28] R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time-dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146. [29] R. B. Salako, W. Shen and S. Xue, Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490.  doi: 10.1007/s00285-019-01400-0. [30] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Eq., 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019. [31] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [32] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [33] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [34] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Part. Differential Eq., 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [35] 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. [36] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057. [37] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [38] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Eq., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [39] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [40] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., (2015), 1125–1133. doi: 10.3934/proc.2015.1125. [41] J. Zheng, Y. Y. Li, G. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064. [42] P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031.
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