February  2021, 41(2): 813-847. doi: 10.3934/dcds.2020301

Asymptotic dynamics of a system of conservation laws from chemotaxis

1. 

Department of Mathematics, Nanchang University, Nanchang 330031, China

2. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

3. 

School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China

4. 

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

* Corresponding author: Kun Zhao

Received  August 2019 Revised  January 2020 Published  August 2020

This paper is devoted to the analytical study of the long-time asymptotic behavior of solutions to the Cauchy problem of a system of conservation laws in one space dimension, which is derived from a repulsive chemotaxis model with singular sensitivity and nonlinear chemical production rate. Assuming the $ H^2 $-norm of the initial perturbation around a constant ground state is finite and using energy methods, we show that there exists a unique global-in-time solution to the Cauchy problem, and the constant ground state is globally asymptotically stable. In addition, the explicit decay rates of the solutions to the chemically diffusive and non-diffusive models are identified under different exponent ranges of the nonlinear chemical production function.

Citation: Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
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[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588–1597. doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

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.  Google Scholar

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C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311–1332. doi: 10.1016/j.jde.2014.05.014.  Google Scholar

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J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687–695. doi: 10.1016/j.jmaa.2012.05.036.  Google Scholar

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M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330–1355. doi: 10.1137/S0036141001385046.  Google Scholar

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

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J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629–641. doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar

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C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew. Math. Phys., 63 (2012), 825–834. doi: 10.1007/s00033-012-0193-0.  Google Scholar

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T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183–217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

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

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Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case, SIAM J. Math. Anal., 50 (2018), 3058–3091. doi: 10.1137/17M112748X.  Google Scholar

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H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193–219. doi: 10.1016/j.jde.2013.04.002.  Google Scholar

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

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E.F. Keller, L.A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

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H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683–730. doi: 10.1137/S0036139995291106.  Google Scholar

[21]

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[22]

D. Li, R. Pan and K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181–2210. doi: 10.1088/0951-7715/28/7/2181.  Google Scholar

[23]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic conservation laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302–338. doi: 10.1016/j.jde.2014.09.014.  Google Scholar

[24]

J. Li, T. Li and Z. 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]

T. Li, R. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417–443. doi: 10.1137/110829453.  Google Scholar

[26]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522–1541. doi: 10.1137/09075161X.  Google Scholar

[27]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967–1998. doi: 10.1142/S0218202510004830.  Google Scholar

[28]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310–1333. doi: 10.1016/j.jde.2010.09.020.  Google Scholar

[29]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161–168. doi: 10.1016/j.mbs.2012.07.003.  Google Scholar

[30]

V. Martinez, Z. Wang and K. Zhao, Asymptotic and viscous stability of large amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383–1424. doi: 10.1512/iumj.2018.67.7394.  Google Scholar

[31]

M. Mei, H. Peng and Z. Wang, Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168–5191. doi: 10.1016/j.jde.2015.06.022.  Google Scholar

[32]

J. D. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[33]

H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044–1081.  Google Scholar

[34]

H. Peng and Z. Wang, Nonlinear stability of strong traveling waves for the singular Keller-Segel system with large perturbations, J. Differential Equations, 265 (2018), 2577–2613. doi: 10.1016/j.jde.2018.04.041.  Google Scholar

[35]

H. Peng, Z. Wang, K. Zhao and C. Zhu, Boundary layers and stabilization of the singular Keller-Segel system, Kinetic and Related Models, 11 (2018), 1085–1123. doi: 10.3934/krm.2018042.  Google Scholar

[36]

H. Peng, H. Wen and C. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis., Z. Angew. Math. Phys., 65 (2014), 1167–1188. doi: 10.1007/s00033-013-0378-1.  Google Scholar

[37]

L. RebholzD. WangZ. WangC. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Disc. Cont. Dyn. Syst. Ser. A, 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154.  Google Scholar

[38]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705–2722. doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar

[39]

Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821–845. doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[40]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1–36. doi: 10.1142/S0218202512500443.  Google Scholar

[41]

D. Wang, Z. Wang and K. Zhao, Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model in multi-dimensions, Indiana Univ. Math. J., in press. Google Scholar

[42]

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

[43]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45–70. doi: 10.1002/mma.898.  Google Scholar

[44]

Z. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225–2258. doi: 10.1016/j.jde.2015.09.063.  Google Scholar

[45]

Z. Wang and K. Zhao, Global dynamics and diffusion limit of a parabolic system arising from repulsive chemotaxis, Commun. Pure Appl. Anal., 12 (2013), 3027–3046. doi: 10.3934/cpaa.2013.12.3027.  Google Scholar

[46]

Y. Zeng, Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyper. Differ. Equ., 14 (2017), 359–391. doi: 10.1142/S0219891617500126.  Google Scholar

[47]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2006), 1017–1027. doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

[48]

Y. Zhang, Z. Tan and M. Sun, Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system, Nonlinear Anal.: Real World Appl., 14 (2013), 465–482. doi: 10.1016/j.nonrwa.2012.07.009.  Google Scholar

[49]

N. Zhu, Z. Liu, V. Martinez and K. Zhao, Global Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model, SIAM J. Math. Anal., 50 (2018), 5380–5425. doi: 10.1137/17M1135645.  Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708–716. Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588–1597. doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

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.  Google Scholar

[4]

C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311–1332. doi: 10.1016/j.jde.2014.05.014.  Google Scholar

[5]

J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687–695. doi: 10.1016/j.jmaa.2012.05.036.  Google Scholar

[6]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330–1355. doi: 10.1137/S0036141001385046.  Google Scholar

[7]

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

[8]

J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629–641. doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar

[9]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew. Math. Phys., 63 (2012), 825–834. doi: 10.1007/s00033-012-0193-0.  Google Scholar

[10]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183–217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[11]

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

[12]

Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case, SIAM J. Math. Anal., 50 (2018), 3058–3091. doi: 10.1137/17M112748X.  Google Scholar

[13]

Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures. Appl., 130 (2019), 251–287. doi: 10.1016/j.matpur.2019.01.008.  Google Scholar

[14]

Q. Hou, Z. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035–5070. doi: 10.1016/j.jde.2016.07.018.  Google Scholar

[15]

H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193–219. doi: 10.1016/j.jde.2013.04.002.  Google Scholar

[16]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1983. Google Scholar

[17]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[18]

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

[19]

E.F. Keller, L.A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[20]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683–730. doi: 10.1137/S0036139995291106.  Google Scholar

[21]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631–1650. doi: 10.1142/S0218202511005519.  Google Scholar

[22]

D. Li, R. Pan and K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181–2210. doi: 10.1088/0951-7715/28/7/2181.  Google Scholar

[23]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic conservation laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302–338. doi: 10.1016/j.jde.2014.09.014.  Google Scholar

[24]

J. Li, T. Li and Z. 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]

T. Li, R. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417–443. doi: 10.1137/110829453.  Google Scholar

[26]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522–1541. doi: 10.1137/09075161X.  Google Scholar

[27]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967–1998. doi: 10.1142/S0218202510004830.  Google Scholar

[28]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310–1333. doi: 10.1016/j.jde.2010.09.020.  Google Scholar

[29]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161–168. doi: 10.1016/j.mbs.2012.07.003.  Google Scholar

[30]

V. Martinez, Z. Wang and K. Zhao, Asymptotic and viscous stability of large amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383–1424. doi: 10.1512/iumj.2018.67.7394.  Google Scholar

[31]

M. Mei, H. Peng and Z. Wang, Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168–5191. doi: 10.1016/j.jde.2015.06.022.  Google Scholar

[32]

J. D. Murray, Mathematical Biology I: An Introduction, 3rd edition, Springer-Verlag, New York, 2002.  Google Scholar

[33]

H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044–1081.  Google Scholar

[34]

H. Peng and Z. Wang, Nonlinear stability of strong traveling waves for the singular Keller-Segel system with large perturbations, J. Differential Equations, 265 (2018), 2577–2613. doi: 10.1016/j.jde.2018.04.041.  Google Scholar

[35]

H. Peng, Z. Wang, K. Zhao and C. Zhu, Boundary layers and stabilization of the singular Keller-Segel system, Kinetic and Related Models, 11 (2018), 1085–1123. doi: 10.3934/krm.2018042.  Google Scholar

[36]

H. Peng, H. Wen and C. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis., Z. Angew. Math. Phys., 65 (2014), 1167–1188. doi: 10.1007/s00033-013-0378-1.  Google Scholar

[37]

L. RebholzD. WangZ. WangC. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Disc. Cont. Dyn. Syst. Ser. A, 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154.  Google Scholar

[38]

Y. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705–2722. doi: 10.3934/dcdsb.2013.18.2705.  Google Scholar

[39]

Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821–845. doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[40]

Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1–36. doi: 10.1142/S0218202512500443.  Google Scholar

[41]

D. Wang, Z. Wang and K. Zhao, Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model in multi-dimensions, Indiana Univ. Math. J., in press. Google Scholar

[42]

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

[43]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45–70. doi: 10.1002/mma.898.  Google Scholar

[44]

Z. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225–2258. doi: 10.1016/j.jde.2015.09.063.  Google Scholar

[45]

Z. Wang and K. Zhao, Global dynamics and diffusion limit of a parabolic system arising from repulsive chemotaxis, Commun. Pure Appl. Anal., 12 (2013), 3027–3046. doi: 10.3934/cpaa.2013.12.3027.  Google Scholar

[46]

Y. Zeng, Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyper. Differ. Equ., 14 (2017), 359–391. doi: 10.1142/S0219891617500126.  Google Scholar

[47]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2006), 1017–1027. doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

[48]

Y. Zhang, Z. Tan and M. Sun, Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system, Nonlinear Anal.: Real World Appl., 14 (2013), 465–482. doi: 10.1016/j.nonrwa.2012.07.009.  Google Scholar

[49]

N. Zhu, Z. Liu, V. Martinez and K. Zhao, Global Cauchy problem of a system of parabolic conservation laws arising from a Keller-Segel type chemotaxis model, SIAM J. Math. Anal., 50 (2018), 5380–5425. doi: 10.1137/17M1135645.  Google Scholar

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