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} $.
Citation: |
[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.![]() ![]() ![]() |