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On the dynamics of 3D electrified falling films
Global generalized solutions to a chemotaxis model of capillary-sprout growth during tumor angiogenesis
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, P. R. China |
$ \begin{equation*} \begin{split} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot (u\nabla v) + \nabla \cdot (u\nabla w),}&{x \in \Omega ,t > 0,} \\ {{v_t} = \Delta v + \nabla \cdot (v\nabla w) - v + u,}&{x \in \Omega ,t > 0,} \\ {0 = \Delta w - w + u,}&{x \in \Omega ,t > 0,} \end{array}} \right. \end{split} \end{equation*} $ |
$ \int_0^T\int_{\Omega}{\frac{{{u_\varepsilon }v_\varepsilon^p}}{{1 +\varepsilon {u_\varepsilon }}}} $ |
$ p>1 $ |
$ \int_0^T \int_\Omega {\frac{{{u_\varepsilon }}}{{1+\varepsilon {u_\varepsilon }}}\ln^{k}({u_\varepsilon } + 1)}dxdt $ |
$ k\in(1,2) $ |
References:
[1] |
X. Bai and S. Liu,
A new criterion to a two-chemical substances chemotaxis system with critical dimension, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3717-3721.
doi: 10.3934/dcdsb.2018074. |
[2] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[3] |
M. A. J. Chaplain and H. M. Byrne,
Mathematical modelling of wound healing and tumour growth: Two sides of the same coin, Wounds, 8 (1996), 42-48.
|
[4] |
X. Chen, A. Jüngel and J.-G. Liu,
A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.
doi: 10.1007/s10440-013-9858-8. |
[5] |
T. Cieślak, P. Laurençot and C. Morales-Rodrigo,
Global existence and convergence to steady states in a chemorepulsion system, Polish Acad. Sci. Inst. Math., Warsaw, 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.
![]() |
[7] |
K. Fujie and S. Ishida,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.
doi: 10.1016/j.jde.2017.02.031. |
[8] |
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.
|
[9] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for Chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[12] |
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. |
[13] |
H.-Y. Jin and Z.-A. Wang,
Boundedness, blowup and critical mass phenomenon incompeting chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[14] |
H.-Y. Jin and Z.-A. Wang,
Global dynamics of the attraction-repulsion Keller-Segel model in one dimension, Math. Methods Appl. Sci., 38 (2015), 444-457.
|
[15] |
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. |
[16] |
P. Laurençot,
Global bounded and unbounded solutions to a chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6419-6444.
|
[17] |
G. Li and Y. Tao, Analysis of a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis, J. Math. Anal. Appl., 481 (2020), 123474, 14 pp.
doi: 10.1016/j.jmaa.2019.123474. |
[18] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[19] |
P. Liu, J. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[20] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
|
[21] |
M. E. Orme and M. A. J. Chaplain,
A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, J. Math. Appl. Med. Biol., 13 (1996), 73-98.
doi: 10.1093/imammb/13.2.73. |
[22] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[23] |
N. Paweletz and M. Knierim,
Tumor related angiogenesis, Crit. Rev. Oncol. Hematol., 9 (1989), 197-242.
doi: 10.1016/S1040-8428(89)80002-2. |
[24] |
M. M. Sholley, G. P. Ferguson, H. R. Seibel, J. L. Montour and J. D. Wilson,
Mechanisms of neovascularization: Vascular sprouting can occur without proliferation of endothelial cells, Lab. Invest., 51 (1984), 624-634.
|
[25] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM. J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[26] |
R. M. Sutherland,
Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.
|
[27] |
Y. 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. |
[28] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[29] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[30] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[31] |
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. |
[32] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[33] |
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. |
[34] |
M. Winkler,
Global mass-preserving solutions in a two-dimensional chemotaxis-stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.
doi: 10.1007/s00028-018-0440-8. |
[35] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM. J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[36] |
M. Winkler,
Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.
doi: 10.1142/S021820251950012X. |
show all references
References:
[1] |
X. Bai and S. Liu,
A new criterion to a two-chemical substances chemotaxis system with critical dimension, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3717-3721.
doi: 10.3934/dcdsb.2018074. |
[2] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[3] |
M. A. J. Chaplain and H. M. Byrne,
Mathematical modelling of wound healing and tumour growth: Two sides of the same coin, Wounds, 8 (1996), 42-48.
|
[4] |
X. Chen, A. Jüngel and J.-G. Liu,
A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.
doi: 10.1007/s10440-013-9858-8. |
[5] |
T. Cieślak, P. Laurençot and C. Morales-Rodrigo,
Global existence and convergence to steady states in a chemorepulsion system, Polish Acad. Sci. Inst. Math., Warsaw, 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.
![]() |
[7] |
K. Fujie and S. Ishida,
Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.
doi: 10.1016/j.jde.2017.02.031. |
[8] |
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.
|
[9] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for Chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
B. Hu and Y. Tao,
To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[12] |
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. |
[13] |
H.-Y. Jin and Z.-A. Wang,
Boundedness, blowup and critical mass phenomenon incompeting chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[14] |
H.-Y. Jin and Z.-A. Wang,
Global dynamics of the attraction-repulsion Keller-Segel model in one dimension, Math. Methods Appl. Sci., 38 (2015), 444-457.
|
[15] |
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. |
[16] |
P. Laurençot,
Global bounded and unbounded solutions to a chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6419-6444.
|
[17] |
G. Li and Y. Tao, Analysis of a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis, J. Math. Anal. Appl., 481 (2020), 123474, 14 pp.
doi: 10.1016/j.jmaa.2019.123474. |
[18] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[19] |
P. Liu, J. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[20] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.
|
[21] |
M. E. Orme and M. A. J. Chaplain,
A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, J. Math. Appl. Med. Biol., 13 (1996), 73-98.
doi: 10.1093/imammb/13.2.73. |
[22] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[23] |
N. Paweletz and M. Knierim,
Tumor related angiogenesis, Crit. Rev. Oncol. Hematol., 9 (1989), 197-242.
doi: 10.1016/S1040-8428(89)80002-2. |
[24] |
M. M. Sholley, G. P. Ferguson, H. R. Seibel, J. L. Montour and J. D. Wilson,
Mechanisms of neovascularization: Vascular sprouting can occur without proliferation of endothelial cells, Lab. Invest., 51 (1984), 624-634.
|
[25] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM. J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[26] |
R. M. Sutherland,
Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.
|
[27] |
Y. 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. |
[28] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[29] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[30] |
J. I. Tello and D. Wrzosek,
Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.
doi: 10.1142/S0218202516400108. |
[31] |
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. |
[32] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[33] |
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. |
[34] |
M. Winkler,
Global mass-preserving solutions in a two-dimensional chemotaxis-stokes system with rotational flux components, J. Evol. Equ., 18 (2018), 1267-1289.
doi: 10.1007/s00028-018-0440-8. |
[35] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM. J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[36] |
M. Winkler,
Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.
doi: 10.1142/S021820251950012X. |
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