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On the scale dynamics of the tropical cyclone intensity
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions
1. | Department of Mathematics, South China University of Technology, Guangzhou 510640, China |
2. | Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China |
$\begin{equation*}\begin{cases}u_t=\nabla· (D(u) \nabla u)-χ\nabla·( u\nabla v)+ξ\nabla·( u\nabla w), &x∈ Ω, ~~t>0,\\ v_t=Δ v+α u-β v,& x∈ Ω, ~~t>0,\\0=Δ w+γ u-δ w, &x∈ Ω, ~~t>0,\\u(x,0)=u_0(x),~v(x,0)=v_0(x),&x∈ Ω,\end{cases}\end{equation*}$ |
$χ, ξ, α, β, γ$ |
$δ$ |
$D(u)$ |
$D(u)≥ d u^θ, u>0$ |
$d>0, θ∈\mathbb{R}$ |
$θ>1-\frac{2}{n}$ |
$\{ξγ> χα\}$ |
$\{ξγ=χα, β≥ δ\}$ |
$θ>1-\frac{4}{n+2}$ |
$θ>1-\frac{2}{n}$ |
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅱ, Commun. Pure Appl.
Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[4] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[5] |
A. Blanchet, J. A. Carrillo and Ph. Laurencǫt,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
doi: 10.1007/s00526-008-0200-7. |
[6] |
T. Cieślak, Ph. Laurencǫt and C. Morales-Rodrigo,
Global existence and convergence to steady states in a chemorepulsion system, equations, Parabolic and Navier-Stokes Equations, in: Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[7] |
T. Cieślak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[8] |
T. Cieślak and C. Stinner,
New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.
doi: 10.1016/j.jde.2014.12.004. |
[9] |
E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[10] |
A. Friedman,
Partial Differential Equations Holt, Rinehart Winston, New York, 1969. |
[11] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[12] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165.
|
[13] |
S. Ishida, T. Ono and T. Yokota,
Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Math. Methods Appl. Sci., 36 (2013), 745-760.
doi: 10.1002/mma.2622. |
[14] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[15] |
S. Ishida and T. Yokota,
Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440.
doi: 10.1016/j.jde.2011.02.012. |
[16] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate KellerSegel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
doi: 10.3934/dcdsb.2013.18.2569. |
[17] |
K. Ishige, Ph. Laurençot and N. Mizoguchi,
Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller-Segel system, Math. Ann., 367 (2017), 461-499.
doi: 10.1007/s00208-016-1400-7. |
[18] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[19] |
H. Y. Jin and Z. Liu,
Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.
doi: 10.1016/j.aml.2015.03.004. |
[20] |
H. Y. Jin and Z. A. Wang,
Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.
doi: 10.1002/mma.3080. |
[21] |
H. Y. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[22] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[23] |
Y. Li and Y. X. Li,
Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.
doi: 10.1016/j.nonrwa.2015.12.003. |
[24] |
K. Lin and C. Mu,
Global existence and convergence to steady states for an attractionrepulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642.
doi: 10.1016/j.nonrwa.2016.03.012. |
[25] |
K. Lin, C. Mu and Y. Gao,
Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572.
doi: 10.1016/j.jde.2016.07.002. |
[26] |
D. Liu and Y. S. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[27] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn. Suppl.1, 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[28] |
P. Liu, J. P. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[29] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signalling,
Microglia, and Alzheimer's disease senile plagues: is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
|
[30] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math.Sci. Appl., 5 (1995), 581-601.
|
[31] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. Ser. Internat., 40 (1997), 411-433.
|
[32] |
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. |
[33] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737.
|
[34] |
K. J. Painter and T. Hillen,
Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[35] |
R. Shi and W. Wang,
Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.
doi: 10.1016/j.jmaa.2014.10.006. |
[36] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[37] |
Y. S. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[38] |
Y. S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math.
Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[39] |
Y. S. 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. |
[40] |
M. Winkler,
A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[41] |
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. |
[42] |
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. |
[43] |
H. Yu, Q. Guo and S. Zheng,
Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342.
doi: 10.1016/j.nonrwa.2016.09.007. |
show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅱ, Commun. Pure Appl.
Math., 17 (1964), 35-92.
doi: 10.1002/cpa.3160170104. |
[3] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[4] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[5] |
A. Blanchet, J. A. Carrillo and Ph. Laurencǫt,
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168.
doi: 10.1007/s00526-008-0200-7. |
[6] |
T. Cieślak, Ph. Laurencǫt and C. Morales-Rodrigo,
Global existence and convergence to steady states in a chemorepulsion system, equations, Parabolic and Navier-Stokes Equations, in: Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[7] |
T. Cieślak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[8] |
T. Cieślak and C. Stinner,
New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.
doi: 10.1016/j.jde.2014.12.004. |
[9] |
E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[10] |
A. Friedman,
Partial Differential Equations Holt, Rinehart Winston, New York, 1969. |
[11] |
K. Fujie, A. Ito, M. Winkler and T. Yokota,
Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.
doi: 10.3934/dcds.2016.36.151. |
[12] |
D. Horstmann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165.
|
[13] |
S. Ishida, T. Ono and T. Yokota,
Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Math. Methods Appl. Sci., 36 (2013), 745-760.
doi: 10.1002/mma.2622. |
[14] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[15] |
S. Ishida and T. Yokota,
Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440.
doi: 10.1016/j.jde.2011.02.012. |
[16] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate KellerSegel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
doi: 10.3934/dcdsb.2013.18.2569. |
[17] |
K. Ishige, Ph. Laurençot and N. Mizoguchi,
Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller-Segel system, Math. Ann., 367 (2017), 461-499.
doi: 10.1007/s00208-016-1400-7. |
[18] |
H. Y. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[19] |
H. Y. Jin and Z. Liu,
Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.
doi: 10.1016/j.aml.2015.03.004. |
[20] |
H. Y. Jin and Z. A. Wang,
Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.
doi: 10.1002/mma.3080. |
[21] |
H. Y. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[22] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[23] |
Y. Li and Y. X. Li,
Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.
doi: 10.1016/j.nonrwa.2015.12.003. |
[24] |
K. Lin and C. Mu,
Global existence and convergence to steady states for an attractionrepulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642.
doi: 10.1016/j.nonrwa.2016.03.012. |
[25] |
K. Lin, C. Mu and Y. Gao,
Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572.
doi: 10.1016/j.jde.2016.07.002. |
[26] |
D. Liu and Y. S. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[27] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn. Suppl.1, 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[28] |
P. Liu, J. P. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[29] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signalling,
Microglia, and Alzheimer's disease senile plagues: is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
|
[30] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math.Sci. Appl., 5 (1995), 581-601.
|
[31] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. Ser. Internat., 40 (1997), 411-433.
|
[32] |
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. |
[33] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737.
|
[34] |
K. J. Painter and T. Hillen,
Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[35] |
R. Shi and W. Wang,
Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.
doi: 10.1016/j.jmaa.2014.10.006. |
[36] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[37] |
Y. S. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[38] |
Y. S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math.
Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[39] |
Y. S. 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. |
[40] |
M. Winkler,
A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[41] |
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. |
[42] |
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. |
[43] |
H. Yu, Q. Guo and S. Zheng,
Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342.
doi: 10.1016/j.nonrwa.2016.09.007. |
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Jianing Xie. Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4007-4022. doi: 10.3934/dcdsb.2021216 |
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