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Recurrent solutions of the Schrödinger-KdV system with boundary forces
Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces
College of Information and Management Science, Henan Agricultural University, Zhengzhou, Henan 450002, China |
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) $.
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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.
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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.
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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.
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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.
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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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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. |
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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. |
[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. |
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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