-
Previous Article
Liouville-type theorem for higher-order Hardy-Hénon system
- CPAA Home
- This Issue
-
Next Article
Continuous solution for a non-linear eikonal system
Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production
1. | College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
3. | School of Sciences, Southwest Petroleum University, Chengdu 610500, China |
4. | College of Mathematics and Statistics, Yili Normal University, Yining 835000, China |
$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*} $ |
$ B_R = B_R(0)\subset \mathbb{R}^n $ |
$ n\geq 1 $ |
$ R>0 $ |
$ \chi>0, \kappa>0 $ |
$ \chi, \kappa $ |
$ \int_{B_R}u_{0} $ |
References:
[1] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: maximally extended solutions and absence of gradient blow-up, Commun. Partial Differ. Equ., 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[2] |
N. Bellomo and M. Winkler,
Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.
doi: 10.1090/btran/17. |
[3] |
A. Chertock, A. Kurganov, X.F. Wang and Y. P. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[4] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1892-1904.
doi: 10.3934/dcds.2015.35.1891. |
[5] |
Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta Appl. Math., (2019), 1-29.
doi: 10.1007/s10440-019-00275-z. |
[6] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[7] |
K. Kanga and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[8] |
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. |
[9] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., 23 (1968). |
[10] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[11] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[12] |
Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 123376.
doi: 10.1016/j.jmaa.2019.123376. |
[13] |
M. Mizukami, T. Ono and T. Yokota,
Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system, J. Differ. Equ., 267 (2019), 5115-5164.
doi: 10.1016/j.jde.2019.05.026. |
[14] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray,
Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.
|
[15] |
M. R. Myerscough, P. K. Maini and K. J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
|
[16] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[17] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[18] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.
|
[19] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[20] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[21] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differ. Integral Equ., 19 (2006), 841-876.
|
[22] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[23] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[24] |
M. Winkler, How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases, Math. Ann., 373 (2019), 1237-1282.
doi: 10.1007/s00208-018-1722-8. |
[25] |
M. Winkler and K. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
show all references
References:
[1] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: maximally extended solutions and absence of gradient blow-up, Commun. Partial Differ. Equ., 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[2] |
N. Bellomo and M. Winkler,
Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.
doi: 10.1090/btran/17. |
[3] |
A. Chertock, A. Kurganov, X.F. Wang and Y. P. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[4] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1892-1904.
doi: 10.3934/dcds.2015.35.1891. |
[5] |
Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta Appl. Math., (2019), 1-29.
doi: 10.1007/s10440-019-00275-z. |
[6] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[7] |
K. Kanga and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[8] |
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. |
[9] |
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., 23 (1968). |
[10] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[11] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[12] |
Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 123376.
doi: 10.1016/j.jmaa.2019.123376. |
[13] |
M. Mizukami, T. Ono and T. Yokota,
Extensibility criterion ruling out gradient blow-up in a quasilinear degenerate chemotaxis system, J. Differ. Equ., 267 (2019), 5115-5164.
doi: 10.1016/j.jde.2019.05.026. |
[14] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray,
Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.
|
[15] |
M. R. Myerscough, P. K. Maini and K. J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
|
[16] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[17] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[18] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.
|
[19] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[20] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[21] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differ. Integral Equ., 19 (2006), 841-876.
|
[22] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[23] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[24] |
M. Winkler, How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases, Math. Ann., 373 (2019), 1237-1282.
doi: 10.1007/s00208-018-1722-8. |
[25] |
M. Winkler and K. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[1] |
J. Ignacio Tello. Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022045 |
[2] |
Hengling Wang, Yuxiang Li. Boundedness in prey-taxis system with rotational flux terms. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4839-4851. doi: 10.3934/cpaa.2020214 |
[3] |
Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 |
[4] |
Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789 |
[5] |
Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147 |
[6] |
Xie Li, Yilong Wang. Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2717-2729. doi: 10.3934/dcdsb.2017132 |
[7] |
Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 |
[8] |
Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737 |
[9] |
Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 191-221. doi: 10.3934/dcdsb.2019178 |
[10] |
Xiangdong Zhao. Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5095-5100. doi: 10.3934/dcdsb.2020334 |
[11] |
Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, 2021, 29 (5) : 3261-3279. doi: 10.3934/era.2021037 |
[12] |
Aichao Liu, Binxiang Dai, Yuming Chen. Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021306 |
[13] |
Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064 |
[14] |
Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic and Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51 |
[15] |
Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066 |
[16] |
Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 |
[17] |
Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031 |
[18] |
Guoqiang Ren, Bin Liu. Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3843-3883. doi: 10.3934/cpaa.2020170 |
[19] |
Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299 |
[20] |
Liangchen Wang, Chunlai Mu. A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4585-4601. doi: 10.3934/dcdsb.2020114 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]