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November  2021, 20(11): 3825-3849. doi: 10.3934/cpaa.2021133

Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production

Hong Yi 1,, , Chunlai Mu 2, , Shuyan Qiu 3, and  Lu Xu 4,

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

* Corresponding author

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

Fund Project: This work is supported in part by the NSFC under grants 11771062 and 11971082, the Fundamental Research Funds for the Central Universities under grants 2019CDJCYJ001, 2020CDJQY-Z001, Chongqing Key Laboratory of Analytic Mathematics and Applications, and Scientific Research Program of the Higher Education Institution of XinJiang under grant XJEDU2021Y043

The following degenerate chemotaxis system with flux limitation and nonlinear signal production
$ \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*} $
is considered in balls
$ B_R = B_R(0)\subset \mathbb{R}^n $
 for
$ n\geq 1 $
 and
$ R>0 $
 with no-flux boundary conditions, where
$ \chi>0, \kappa>0 $
. We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for
$ \chi, \kappa $
 and
$ \int_{B_R}u_{0} $
.
Keywords: Degenerate chemotaxis system, flux limitation, boundedness.
Mathematics Subject Classification: Primary: 35K65, 35B51; Secondary: 39A22.
Citation: Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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