September  2021, 41(9): 4065-4083. doi: 10.3934/dcds.2021028

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

* Corresponding author: Xueli Bai

Received  August 2020 Revised  December 2020 Published  January 2021

This paper considers a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis
$ \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*} $
under Neumann initial-boundary conditions in a smooth bounded domain. In the two-dimensional setting, introducing a generalized solution concept according to (Winkler, 2015 [35]) and constructing an appropriate regularized system, we prove the global existence of at least one such solution with suitably regular initial data by an approximation procedure. To overcome the difficulty in taking the limit to its regularized system, we establish some technical estimates related to several energy integrals with special structures like
$ \int_0^T\int_{\Omega}{\frac{{{u_\varepsilon }v_\varepsilon^p}}{{1 +\varepsilon {u_\varepsilon }}}} $
,
$ p>1 $
and
$ \int_0^T \int_\Omega {\frac{{{u_\varepsilon }}}{{1+\varepsilon {u_\varepsilon }}}\ln^{k}({u_\varepsilon } + 1)}dxdt $
,
$ k\in(1,2) $
.
Citation: Xueli Bai, Wenji Zhang. Global generalized solutions to a chemotaxis model of capillary-sprout growth during tumor angiogenesis. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4065-4083. doi: 10.3934/dcds.2021028
References:
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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.  Google Scholar

[2]

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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P. LiuJ. 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.  Google Scholar

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M. LucaA. Chavez-RossL. 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.   Google Scholar

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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.  Google Scholar

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

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N. Paweletz and M. Knierim, Tumor related angiogenesis, Crit. Rev. Oncol. Hematol., 9 (1989), 197-242.  doi: 10.1016/S1040-8428(89)80002-2.  Google Scholar

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M. M. SholleyG. P. FergusonH. R. SeibelJ. L. Montour and J. D. Wilson, Mechanisms of neovascularization: Vascular sprouting can occur without proliferation of endothelial cells, Lab. Invest., 51 (1984), 624-634.   Google Scholar

[25]

C. StinnerC. 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.  Google Scholar

[26]

R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

X. ChenA. 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.  Google Scholar

[5]

T. CieślakP. 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.  Google Scholar

[6] M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004.   Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P. LiuJ. 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.  Google Scholar

[20]

M. LucaA. Chavez-RossL. 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.   Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

M. M. SholleyG. P. FergusonH. R. SeibelJ. L. Montour and J. D. Wilson, Mechanisms of neovascularization: Vascular sprouting can occur without proliferation of endothelial cells, Lab. Invest., 51 (1984), 624-634.   Google Scholar

[25]

C. StinnerC. 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.  Google Scholar

[26]

R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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