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December  2021, 26(12): 6057-6068. doi: 10.3934/dcdsb.2021002

Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces

Xing Wu , and  Keqin Su

College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, China

* Corresponding author: Xing Wu

Received  August 2020 Revised  November 2020 Published  December 2021 Early access  December 2020

Fund Project: This work is partially supported by NSF of China (No.11801090)

In this paper, we study the qualitative behavior of hyperbolic system arising from chemotaxis models. Firstly, by establishing a new product estimates in multi-dimensional Besov space $ \dot{B}_{2, r}^{\frac d2}(\mathbb{R}^d)(1\leq r\leq \infty) $, we establish the global small solutions in multi-dimensional Besov space $ \dot{B}_{2, r}^{\frac d2-1}(\mathbb{R}^d) $ by the method of energy estimates. Then we study the asymptotic behavior and obtain the optimal decay rate of the global solutions if the initial data are small in $ B_{2, 1}^{\frac{d}{2}-1}(\mathbb{R}^d)\cap \dot{B}_{1, \infty}^0(\mathbb{R}^d) $.

Keywords: Chemotaxis system, global small solutions, optimal decay rate.
Mathematics Subject Classification: Primary:92C17, 35L52;Secondary:35B40.
Citation: Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6057-6068. doi: 10.3934/dcdsb.2021002
References:
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H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

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P. Biler, G. Karch and J. Zienkiewicz, Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math., 330 (2018), 834-875.  doi: 10.1016/j.aim.2018.03.036.

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

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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.

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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.

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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.

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D. Hortsmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165. 

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J. Li, T. 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.

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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.

[12]

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

[13]

D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbb{R}^n$ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

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T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.  doi: 10.1137/09075161X.

[15]

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

[16]

T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equ., 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[17]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differ. Equ., 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.

[18]

V. R. 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.

[19]

M. Okita, Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differ. Equ., 257 (2014), 3850-3867.  doi: 10.1016/j.jde.2014.07.011.

[20]

Y. Tao, L. Wang and Z.-A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Continuous Dynam. Systems - B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821.

[21]

Z.-A. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equ., 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[22]

Y. Zhang, Z. Tan and M. B. 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.

[23]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

P. Biler, G. Karch and J. Zienkiewicz, Large global-in-time solutions to a nonlocal model of chemotaxis, Adv. Math., 330 (2018), 834-875.  doi: 10.1016/j.aim.2018.03.036.

[3]

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

[4]

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.

[5]

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.

[6]

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.

[7]

D. Hortsmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein, 105 (2003), 103-165. 

[8]

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.

[9]

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.

[10]

J. Li, T. 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.

[11]

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.

[12]

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

[13]

D. Li and J. Rodrigo, Finite-time singularities of an aggregation equation in $\mathbb{R}^n$ with fractional dissipation, Comm. Math. Phys., 287 (2009), 687-703.  doi: 10.1007/s00220-008-0669-0.

[14]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.  doi: 10.1137/09075161X.

[15]

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

[16]

T. Li and Z.-A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equ., 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[17]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differ. Equ., 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.

[18]

V. R. 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.

[19]

M. Okita, Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differ. Equ., 257 (2014), 3850-3867.  doi: 10.1016/j.jde.2014.07.011.

[20]

Y. Tao, L. Wang and Z.-A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Continuous Dynam. Systems - B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821.

[21]

Z.-A. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equ., 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[22]

Y. Zhang, Z. Tan and M. B. 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.

[23]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

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