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 & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268
References:
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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

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[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[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. Google Scholar

[17]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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