-
Previous Article
Variational solutions of stochastic partial differential equations with cylindrical Lévy noise
- DCDS-B Home
- This Issue
-
Next Article
Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory
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) $.
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. |
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. |
[1] |
Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198 |
[2] |
Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021007 |
[3] |
Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020210 |
[4] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[5] |
Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194 |
[6] |
Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 |
[7] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[8] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[9] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[10] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[11] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[12] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[13] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[14] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[15] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[16] |
Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021012 |
[17] |
Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 |
[18] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[19] |
Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207 |
[20] |
Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]