# American Institute of Mathematical Sciences

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  September 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (9) : 4065-4083. doi: 10.3934/dcds.2021028
##### 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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##### 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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