December  2017, 37(12): 6189-6225. doi: 10.3934/dcds.2017268

Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849, USA

* Corresponding author: Rachidi B. Salako

Received  December 2016 Revised  July 2017 Published  August 2017

The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,
$\label{IntroEq0-2}\begin{cases}u_{t}=Δ{u}-χ\nabla·(u\nabla{v})+u(1-u),{x}∈\mathbb{R}^N,\\{0}=Δ{v}-v+u,{x}∈\mathbb{R}^N,\end{cases}$
where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 < χ < 1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x, t;u_0), v(x, t;u_0))$ with initial condition $u(x, 0;u_0)=u_0(x)$. Furthermore, it was shown that, if $0 < χ < \frac{1}{2}$, then the constant steady-state solution $(1, 1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0 < χ < 1$, then there are nonnegative constants $c_ - ^*\left( \chi \right) \le c_ + ^*\left( \chi \right)$ such that for every nonnegative initial function $u_0(·)$ with non-empty and compact support ${\rm{supp}}(u_0)$,
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [|u(x,t;{u_0}) - 1| + |v(x,t;{u_0}) - 1|] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} 0 < c < c_ - ^*(\chi )$
and
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$
We also show that if $0 < χ < \frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c \ge c^*(χ)$, the system has a traveling wave solution $(u(x, t), v(x, t))$ with speed $c$ and connecting $(1, 1)$ and $(0, 0)$, that is, $(u(x, t), v(x, t))=(U(x-ct), V(x-ct))$ for some functions $U(·)$ and $V(·)$ satisfying $(U(-∞), V(-∞))=(1, 1)$ and $(U(∞), V(∞))=(0, 0)$. Moreover, we show that
$\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$
We then consider the extensions of the results in the case $N=1$ to the case $N \ge 2$.
Citation: Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268
References:
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S. AiW. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21.  doi: 10.3934/dcdsb.2015.20.1.

[2]

S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.  doi: 10.3934/mbe.2015.12.717.

[3]

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.

[4]

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

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅰ -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.

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H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint.

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M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285.

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J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680. 

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J. I. DiazT. 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.

[11]

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

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M. Freidlin, On wave front propagation in periodic media, In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166. 

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M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. 

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A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.

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M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.  doi: 10.4171/IFB/141.

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E. GalakhovO. 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.

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.

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D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.  doi: 10.1007/s00332-003-0548-y.

[19]

K. Kanga, Angela Steven Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.

[20]

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.

[21]

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.

[22]

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. 

[23]

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.

[24]

J. LiT. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389.

[25]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[26]

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.

[27]

B. P. MarchantJ. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.  doi: 10.1088/0951-7715/14/6/313.

[28]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[29]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.  doi: 10.4171/IFB/200.

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. 

[31]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.

[32]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.

[33]

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.

[34]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[35]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating 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.

[36]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. 

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. 

[38]

Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[39]

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.

[40]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.

[41]

L. WangC. 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.

[42]

Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.

[43]

H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[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, 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.

[46]

M. Winkler, Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[47]

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.

[48]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.

[49]

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.

[50]

T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst, (2015), 1125-1133.  doi: 10.3934/proc.2015.1125.

[51]

P. ZhengC. MuX. 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.

[52]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl.(9), 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.

show all references

References:
[1]

S. AiW. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21.  doi: 10.3934/dcdsb.2015.20.1.

[2]

S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.  doi: 10.3934/mbe.2015.12.717.

[3]

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.

[4]

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

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅰ -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.

[6]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.

[7]

H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint.

[8]

M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285.

[9]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680. 

[10]

J. I. DiazT. 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.

[11]

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

[12]

M. Freidlin, On wave front propagation in periodic media, In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166. 

[13]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. 

[14]

A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.

[15]

M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.  doi: 10.4171/IFB/141.

[16]

E. GalakhovO. 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.

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.

[18]

D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.  doi: 10.1007/s00332-003-0548-y.

[19]

K. Kanga, Angela Steven Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.

[20]

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.

[21]

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.

[22]

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. 

[23]

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.

[24]

J. LiT. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389.

[25]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.

[26]

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.

[27]

B. P. MarchantJ. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.  doi: 10.1088/0951-7715/14/6/313.

[28]

G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.

[29]

G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.  doi: 10.4171/IFB/200.

[30]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. 

[31]

J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.

[32]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.

[33]

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.

[34]

D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.

[35]

W. Shen, Variational principle for spatial spreading speeds and generalized propgating 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.

[36]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. 

[37]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. 

[38]

Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.

[39]

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.

[40]

K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.

[41]

L. WangC. 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.

[42]

Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.

[43]

H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.

[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, 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.

[46]

M. Winkler, Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[47]

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.

[48]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.

[49]

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.

[50]

T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst, (2015), 1125-1133.  doi: 10.3934/proc.2015.1125.

[51]

P. ZhengC. MuX. 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.

[52]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl.(9), 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.

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