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  February 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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