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Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA |
$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad \ x\in \mathbb{R} \\ \tau {{\partial }_{t}}{{v}_{1}} = (\Delta -{{\lambda }_{1}}I){{v}_{1}}+{{\mu }_{1}}u,\qquad \ x\in \mathbb{R}, \\ \tau \partial {{v}_{2}} = (\Delta -{{\lambda }_{2}}I){{v}_{2}}+{{\mu }_{2}}u,\qquad \ \ x\in \mathbb{R}, \\\end{array} \right. & (0.1) \\\end{matrix} $ |
$ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $ |
$ i = 1,2 $ |
$ \ a>0,\ b> 0 $ |
$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) $ |
$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2, \lambda_2) $ |
$ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $ |
$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ |
$ (0,0,0) $ |
$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $ |
$ \mu\in (0,\sqrt a) $ |
$ c = c_\mu: = \mu+\frac{a}{\mu} $ |
$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $ |
$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $ |
$ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $ |
$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ |
$ (0,0,0) $ |
$ c<2\sqrt{a} $ |
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] |
H. Berestycki, F. 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. |
[3] |
H. Berestycki, F. Hamel and 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. |
[4] |
H. Berestycki, F. 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. |
[5] |
H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. Google Scholar |
[6] |
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.
|
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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 |
[11] |
A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[12] |
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. |
[13] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. |
[14] |
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. |
[15] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[16] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[17] |
D. Horstmann,
From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.
|
[18] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[19] |
K. Kanga and A. 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. Kolmogorov, I. 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] |
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. |
[24] |
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. |
[25] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[26] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimers disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar |
[27] |
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. |
[28] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser Inequality to a Parabolic System of Chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[29] |
J. Nolen, M. 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. |
[30] |
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. |
[31] |
R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$, Journal of Dynamics and Differential Equations, (2017).
doi: 10.1007/s10884-017-9602-6. |
[32] |
R. B. Salako and W. Shen, Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source, Discrete and Continuous Dynamical System-Series S.
doi: 10.3934/dcdss.2020017. |
[33] |
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 and Continuous Dynamical Systems - Series A, 37 (2017), 6189-6225.
doi: 10.3934/dcds.2017268. |
[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] |
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] |
Y. Wang and Z. Xiang,
Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Dyn. Syst. Ser. B, 21 (2016), 1953-1973.
doi: 10.3934/dcdsb.2016031. |
[41] |
H. F. Weinberger,
Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[42] |
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. |
[43] |
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. |
[44] |
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. |
[45] |
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. |
[46] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[47] |
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. |
[48] |
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. |
[49] |
Q. Zhang and Y. Li,
An attraction-repulsion chemotaxis system with logistic source, ZAMMZ. Angew. Math. Mech., 96 (2016), 570-584.
doi: 10.1002/zamm.201400311. |
[50] |
P. Zheng, C. Mu and X. Hu,
Boundedness in the higher dimensional attractionrepulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202.
doi: 10.1016/j.camwa.2016.08.028. |
[51] |
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. |
[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] |
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] |
H. Berestycki, F. 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. |
[3] |
H. Berestycki, F. Hamel and 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. |
[4] |
H. Berestycki, F. 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. |
[5] |
H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. Google Scholar |
[6] |
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.
|
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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 |
[11] |
A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[12] |
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. |
[13] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. |
[14] |
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. |
[15] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[16] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[17] |
D. Horstmann,
From 1970 until present: The KellerSegel model in chemotaxis and its consequences, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.
|
[18] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[19] |
K. Kanga and A. 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. Kolmogorov, I. 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] |
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. |
[24] |
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. |
[25] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[26] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimers disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar |
[27] |
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. |
[28] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser Inequality to a Parabolic System of Chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[29] |
J. Nolen, M. 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. |
[30] |
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. |
[31] |
R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$, Journal of Dynamics and Differential Equations, (2017).
doi: 10.1007/s10884-017-9602-6. |
[32] |
R. B. Salako and W. Shen, Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source, Discrete and Continuous Dynamical System-Series S.
doi: 10.3934/dcdss.2020017. |
[33] |
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 and Continuous Dynamical Systems - Series A, 37 (2017), 6189-6225.
doi: 10.3934/dcds.2017268. |
[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] |
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] |
Y. Wang and Z. Xiang,
Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Dyn. Syst. Ser. B, 21 (2016), 1953-1973.
doi: 10.3934/dcdsb.2016031. |
[41] |
H. F. Weinberger,
Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[42] |
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. |
[43] |
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. |
[44] |
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. |
[45] |
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. |
[46] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[47] |
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. |
[48] |
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. |
[49] |
Q. Zhang and Y. Li,
An attraction-repulsion chemotaxis system with logistic source, ZAMMZ. Angew. Math. Mech., 96 (2016), 570-584.
doi: 10.1002/zamm.201400311. |
[50] |
P. Zheng, C. Mu and X. Hu,
Boundedness in the higher dimensional attractionrepulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202.
doi: 10.1016/j.camwa.2016.08.028. |
[51] |
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. |
[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. |
[1] |
Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 293-319. doi: 10.3934/dcdss.2020017 |
[2] |
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Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125 |
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