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doi: 10.3934/dcdss.2022074
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Spreading speeds of a parabolic-parabolic chemotaxis model 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: Wenxian Shen (wenxish@auburn.edu)

Dedicated to Professor Georg Hetzer on the occasion of his 75th Birthday

Received  July 2021 Revised  February 2022 Early access March 2022

The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on
$ {{\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} $
where
$ \chi, \ a,\ b,\ \lambda,\ \mu $
are positive constants. Assume
$ b>\frac{N\mu\chi}{4} $
. Among others, it is proved that
$ 2\sqrt{a} $
is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is,
$ \lim\limits_{t\to\infty}\sup\limits_{|x|\geq ct}u(x,t;u_0,v_0) = 0\quad \forall\,\, c>2\sqrt{a} $
and
$ \liminf\limits_{t\to\infty}\inf\limits_{|x|\leq ct}u(x,t;u_0,v_0)>0 \quad \forall\,\, 0<c<2\sqrt{a}. $
where
$ (u(x,t;u_0,v_0), v(x,t;u_0,v_0)) $
is the unique global classical solution of (1) with
$ u(x,0;u_0,v_0) = u_0 $
,
$ v(x,0;u_0,v_0) = v_0 $
, and
$ {\rm supp}(u_0) $
,
$ {\rm supp}(v_0) $
are nonempty and compact. It is well known that
$ 2\sqrt{a} $
is the spreading speed of the following Fisher-KPP equation,
$ u_t = \Delta u+u(a-bu),\quad \forall\,\ x\in{{\mathbb R}}^N. $
Hence, if
$ b>\frac{N\mu\chi}{4} $
, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.
Citation: Wenxian Shen, Shuwen Xue. Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $ \mathbb{R}^{N} $. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022074
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 446 (1975), 5–49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl Math., 171 (2021), 82 pp. doi: 10.1007/s10440-020-00374-2.

[4]

N. BellomoA. BellouquidY. 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.

[5]

H. BerestyckiF. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, Journal of Functional Analysis, 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.

[6]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[7]

L. CorriasM. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Differential Equations, 257 (2014), 1840-1878.  doi: 10.1016/j.jde.2014.05.019.

[8] M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. 
[9]

L. C. F. Ferreira and J. C. Precioso, Existence and asymptotic behavior for the parabolic-parabolic Keller-Segel system with singular data, Nonlinearity, 24 (2011), 1433-1449.  doi: 10.1088/0951-7715/24/5/003.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[11]

M. A. HerreroE. Medina and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268.

[12]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 24 (1997), 633-683. 

[13]

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.

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. 

[15]

D. Horstmann and M. Winkler, Boundedness vs. blow up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[16]

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.

[17]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[18]

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.

[19]

E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[20]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. 

[21]

K. KutoK. OsakiT. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009.

[22]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Eq., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[23]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[24]

K. Lin and C. 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.

[25]

N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.  doi: 10.1007/s00526-012-0558-4.

[26]

T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[27]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[28]

T. NagaiR. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in ${{\mathbb R}}^N$, Funkcialaj Ekvacioj, 46 (2003), 383-407.  doi: 10.1619/fesi.46.383.

[29]

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.

[30]

Y. NaitoT. Suzuki and K. Yoshida, Self-Similar solutions to a parabolic system modeling chemotaxis, J. of Differential Equations, 184 (2002), 386-421.  doi: 10.1006/jdeq.2001.4146.

[31]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[32]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[33]

K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, Journal of Theoretical Biology, 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.

[34]

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.

[35]

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.

[36]

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, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.

[37]

R. B. Salako and W. Shen, Existence of Traveling wave solution of parabolic-parabolic chemotaxis systems, Nonlinear Analysis: Real World Applications, 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[38]

R. B. Salako and W. Shen, Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source, Electron. J. Differential Equations, 2020 (2020), 18 pp.

[39]

R. B. SalakoW. 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.

[40]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.

[41]

W. Shen and S. Xue, Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete and Continuous Dynamical Systems, http://dx.doi.org/10.3934/dcds.2022003.

[42]

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.

[43]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[44]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[45]

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.

[46]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[47]

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.

[48]

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.

[49]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Discrete and Continuous Dynamical Systems, 69, (2018), 40 pp. doi: 10.1007/s00033-018-0935-8.

[50]

J. ZhengY. Y. LiG. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source, Journal of Mathematical Analysis and Applications, 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 446 (1975), 5–49.

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.

[3]

G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl Math., 171 (2021), 82 pp. doi: 10.1007/s10440-020-00374-2.

[4]

N. BellomoA. BellouquidY. 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.

[5]

H. BerestyckiF. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, Journal of Functional Analysis, 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.

[6]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[7]

L. CorriasM. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Differential Equations, 257 (2014), 1840-1878.  doi: 10.1016/j.jde.2014.05.019.

[8] M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. 
[9]

L. C. F. Ferreira and J. C. Precioso, Existence and asymptotic behavior for the parabolic-parabolic Keller-Segel system with singular data, Nonlinearity, 24 (2011), 1433-1449.  doi: 10.1088/0951-7715/24/5/003.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[11]

M. A. HerreroE. Medina and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268.

[12]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 24 (1997), 633-683. 

[13]

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.

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. 

[15]

D. Horstmann and M. Winkler, Boundedness vs. blow up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[16]

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.

[17]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[18]

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.

[19]

E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[20]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. 

[21]

K. KutoK. OsakiT. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D, 241 (2012), 1629-1639.  doi: 10.1016/j.physd.2012.06.009.

[22]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Eq., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[23]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.

[24]

K. Lin and C. 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.

[25]

N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.  doi: 10.1007/s00526-012-0558-4.

[26]

T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[27]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[28]

T. NagaiR. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in ${{\mathbb R}}^N$, Funkcialaj Ekvacioj, 46 (2003), 383-407.  doi: 10.1619/fesi.46.383.

[29]

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.

[30]

Y. NaitoT. Suzuki and K. Yoshida, Self-Similar solutions to a parabolic system modeling chemotaxis, J. of Differential Equations, 184 (2002), 386-421.  doi: 10.1006/jdeq.2001.4146.

[31]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[32]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[33]

K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, Journal of Theoretical Biology, 481 (2019), 162-182.  doi: 10.1016/j.jtbi.2018.06.019.

[34]

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.

[35]

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.

[36]

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, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.

[37]

R. B. Salako and W. Shen, Existence of Traveling wave solution of parabolic-parabolic chemotaxis systems, Nonlinear Analysis: Real World Applications, 42 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.12.004.

[38]

R. B. Salako and W. Shen, Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source, Electron. J. Differential Equations, 2020 (2020), 18 pp.

[39]

R. B. SalakoW. 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.

[40]

W. Shen, Variational principle for spatial spreading speeds and generalized propagating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.

[41]

W. Shen and S. Xue, Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete and Continuous Dynamical Systems, http://dx.doi.org/10.3934/dcds.2022003.

[42]

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.

[43]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.

[44]

H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.

[45]

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.

[46]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.

[47]

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.

[48]

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.

[49]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Discrete and Continuous Dynamical Systems, 69, (2018), 40 pp. doi: 10.1007/s00033-018-0935-8.

[50]

J. ZhengY. Y. LiG. Bao and X. Zou, A new result for global existence and boundedness of solutions to a parabolic-parabolic Keller-Segel system with logistic source, Journal of Mathematical Analysis and Applications, 462 (2018), 1-25.  doi: 10.1016/j.jmaa.2018.01.064.

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