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Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip
$ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity
1. | School of Mathematical Sciences, Peking University, Beijing, 100871, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China |
This paper studies the parabolic-parabolic Keller-Segel system with supercritical sensitivity: $u_{t}=\nabla\cdot(\phi (u) \nabla u)-\nabla \cdot(\varphi(u)\nabla v)$, $v_{t}=\Delta v -v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^n$ $(n\ge2)$, the diffusivity fulfills $\phi(u)\ge a_0(u+1)^{\gamma}$ with $\gamma\ge0$ and $a_0>0$, while the chemotactic sensitivity satisfies $0\le \varphi(u)\le b_0u(u+1)^{\alpha+\gamma-1}$ with $\alpha>\frac{2}{n}$ and $b_0>0$. It is proved that the problem possesses a globally bounded solution for $\frac{4}{n+2}<\alpha<2$, whenever $\|u_0\|_{L^{\frac{n\alpha}{2}}(\Omega)}$ and $\|\nabla v_0\|_{L^{\frac{n\alpha+2\gamma}{2-\alpha}}(\Omega)}$ is sufficiently small. Similarly, the above conclusion still holds for $\alpha>2$ provided that $\|u_{0}\|_{L^{n\alpha-n}(\Omega)}$ and $\|\nabla v_0\|_{L^{\infty}(\Omega)}$ are small enough.
References:
[1] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[2] |
T. Cieślak and C. Morales-Rodrigo,
Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect: Existence and uniqueness of global-in-time solutions, Topol. Methods Nonlinear Anal., 29 (2007), 361-381.
|
[3] |
T. Cieślak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[4] |
T. Cieślak and M. Winkler,
Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.
doi: 10.1016/j.nonrwa.2016.10.002. |
[5] |
T. Cieślak and M. Winkler,
Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.
doi: 10.1016/j.na.2016.04.013. |
[6] |
M. Ding and S. Zheng, $ L^\gamma $-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity, Discrete Contin. Dyn. Syst. Ser. B, Online First, (2018).
doi: 10.3934/dcdsb.2018295. |
[7] |
L. Fan and H. Jin, Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys., 58 (2017), 011503, 22 pp.
doi: 10.1063/1.4974245. |
[8] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[9] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683.
|
[10] |
T. Hillen and K. J. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[13] |
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. |
[14] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. Google Scholar |
[15] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[16] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[18] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.
|
[19] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[20] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543.
|
[21] |
T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., (2006), Art. ID 23061, 21 pp.
doi: 10.1155/AAA/2006/23061. |
[22] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[23] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Meth. Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[24] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[25] |
M. Winkler,
A critical exponent in a degenerate parabolic parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[26] |
M. Winkler,
Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.
doi: 10.1088/1361-6544/aa565b. |
[27] |
M. Winkler and K. C. 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. |
[28] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[29] |
H. Yu, W. Wang and S. Zheng,
Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1317-1327.
doi: 10.3934/dcdsb.2016.21.1317. |
[30] |
H. Yu, W. Wang and S. Zheng,
Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1635-1644.
doi: 10.3934/dcdsb.2017078. |
show all references
References:
[1] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[2] |
T. Cieślak and C. Morales-Rodrigo,
Quasilinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect: Existence and uniqueness of global-in-time solutions, Topol. Methods Nonlinear Anal., 29 (2007), 361-381.
|
[3] |
T. Cieślak and M. Winkler,
Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.
doi: 10.1088/0951-7715/21/5/009. |
[4] |
T. Cieślak and M. Winkler,
Global bounded solutions in a two-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal. Real World Appl., 35 (2017), 1-19.
doi: 10.1016/j.nonrwa.2016.10.002. |
[5] |
T. Cieślak and M. Winkler,
Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity, Nonlinear Anal., 159 (2017), 129-144.
doi: 10.1016/j.na.2016.04.013. |
[6] |
M. Ding and S. Zheng, $ L^\gamma $-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity, Discrete Contin. Dyn. Syst. Ser. B, Online First, (2018).
doi: 10.3934/dcdsb.2018295. |
[7] |
L. Fan and H. Jin, Global existence and asymptotic behavior to a chemotaxis system with consumption of chemoattractant in higher dimensions, J. Math. Phys., 58 (2017), 011503, 22 pp.
doi: 10.1063/1.4974245. |
[8] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[9] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683.
|
[10] |
T. Hillen and K. J. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[13] |
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. |
[14] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. Google Scholar |
[15] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[16] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[18] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.
|
[19] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[20] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart, 10 (2002), 501-543.
|
[21] |
T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., (2006), Art. ID 23061, 21 pp.
doi: 10.1155/AAA/2006/23061. |
[22] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[23] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Meth. Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[24] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[25] |
M. Winkler,
A critical exponent in a degenerate parabolic parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[26] |
M. Winkler,
Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.
doi: 10.1088/1361-6544/aa565b. |
[27] |
M. Winkler and K. C. 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. |
[28] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[29] |
H. Yu, W. Wang and S. Zheng,
Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1317-1327.
doi: 10.3934/dcdsb.2016.21.1317. |
[30] |
H. Yu, W. Wang and S. Zheng,
Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1635-1644.
doi: 10.3934/dcdsb.2017078. |
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