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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 |
$ \mathbb{R}^{N} $ |
$ \begin{equation} \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)\end{equation} $ |
$ \chi, \ a,\ b,\ \lambda,\ \mu $ |
$ N $ |
$ b>\frac{N\chi\mu}{4} $ |
$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $ |
$ u(x,0;u_0, v_0) = u_0(x) $ |
$ v(x,0;u_0, v_0) = v_0(x) $ |
$ u_0(x) $ |
$ v_0(x) $ |
$ b>\frac{N\chi\mu}{4} $ |
$ u_0 $ |
$ (u_0, v_0) $ |
$ u_0 $ |
$ K = K(a,\lambda,N)>\frac{N}{4} $ |
$ b>K \chi\mu $ |
$ \lambda\geq \frac{a}{2} $ |
$ u_0(\cdot) $ |
$ \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. $ |
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. |
show all references
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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2020 Impact Factor: 1.392
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