October  2019, 12(5): 1131-1162. doi: 10.3934/krm.2019043

On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks

a. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

b. 

Department of Mathematics, Jincheng College of Sichuan University, Chengdu 611731, China

Received  January 2019 Revised  March 2019 Published  July 2019

In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.

Citation: Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic & Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043
References:
[1]

G. AlbiM. ArtinaM. Foransier and P. Markowich, Biological transportation networks: Modeling and simulation, Analysis Applications, 14 (2016), 1855-206. doi: 10.1142/S0219530515400059. Google Scholar

[2]

G. Albi, M. Burger, J. Haskovec, P. Markowich and M. Schlottbom, Continuum Modeling of Biological Network Formation, in Active Particles, vol. I. Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond and T. Tamdor), Birkhäuser, Boston, 2017, 1–48. Google Scholar

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H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

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N. Dengler and J. Kang, Vascular patterning and leaf shape, Current Opinion in Plant Biology, 4 (2001), 50-56. doi: 10.1016/S1369-5266(00)00135-7. Google Scholar

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A. EichmannF. Le NobleM. Autiero and P. Carmeliet, Guidance of vascular and neural network formation, Current Opinion in Neurobiology, 15 (2005), 108-115. doi: 10.1016/j.conb.2005.01.008. Google Scholar

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A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in $ \mathbb R^n$, Journal of Differential Equations, 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1. Google Scholar

[7]

J. HaskovecP. Markowich and B. Perthame, Mathematical analysis of a PDE system for biological network formation, Comm. Partial Differential Equations, 40 (2015), 918-956. doi: 10.1080/03605302.2014.968792. Google Scholar

[8]

J. HaskovecP. MarkowichB. Perthame and M. Schlottbomc, Notes on a PDE system for biological network formation, Nonlinear Analysis, 138 (2016), 127-155. doi: 10.1016/j.na.2015.12.018. Google Scholar

[9]

D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Physical Review Letters, 111 (2013), 138701. doi: 10.1103/PhysRevLett.111.138701. Google Scholar

[10]

D. Hu, Optimization, Adaptation, and Initialization of Biological Transport Networks, Notes from lecture, 2014.Google Scholar

[11]

E. Katifori, G. Szollosi and M. Magnasco, Damage and fluctuations induce loops in optimal transport networks, Physical Review Letters, 104 (2010), 048704. doi: 10.1103/PhysRevLett.104.048704. Google Scholar

[12]

H. Lin and Z. Xiang, Global well-posedness for 2D incompressible magneto-micropolar fluid system with partial viscosity, Sci. China Math., in press, (2019), http://engine.scichina.com/doi/10.1007/s11425-018-9427-6. doi: 10.1007/s11425-018-9427-6. Google Scholar

[13]

J. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, Journal of Differential Equations, 264 (2018), 5489-5526. doi: 10.1016/j.jde.2018.01.001. Google Scholar

[14]

R. Malinowski, Understanding of leaf development-the science of complexity, Plants, 2 (2013), 396-415. doi: 10.3390/plants2030396. Google Scholar

[15]

O. Michel and J. Biondi, Morphogenesis of neural networks, Neural Processing Letters, 2 (1995), 9-12. doi: 10.1007/BF02312376. Google Scholar

[16]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with finite depth, Math. Models Methods Appl. Sci., 28 (2018), 869-920. doi: 10.1142/S0218202518500239. Google Scholar

[17]

Y. Peng and Z. Xiang, Global existence and convergence rates to achemotaxis-fluids system with mixed boundary conditions, Journal of Differential Equations, 267 (2019), 1277-1321. doi: 10.1016/j.jde.2019.02.007. Google Scholar

[18]

X. RenJ. WuZ. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020. Google Scholar

[19]

X. RenZ. Xiang and Z. Zhang, Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain, Nonlinearity, 29 (2016), 1257-1291. doi: 10.1088/0951-7715/29/4/1257. Google Scholar

[20]

X. RenZ. Xiang and Z. Zhang, Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion, Sci. China Math., 59 (2016), 1949-1974. doi: 10.1007/s11425-016-5145-2. Google Scholar

[21]

D. Sedmera, Function and form in the developing cardiovascular system, Cardiovascular Research, 91 (2011), 252-259. doi: 10.1093/cvr/cvr062. Google Scholar

[22]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344. Google Scholar

[23]

X. Xu, Life-span of smooth solutions to a PDE system with cubic nonlinearity, preprint, arXiv: 1706.06057.Google Scholar

[24]

X. Xu, Regularity theorems for a biological network formulation model in two space dimensions, Kinetic and Related Models, 11 (2018), 397-408. doi: 10.3934/krm.2018018. Google Scholar

show all references

References:
[1]

G. AlbiM. ArtinaM. Foransier and P. Markowich, Biological transportation networks: Modeling and simulation, Analysis Applications, 14 (2016), 1855-206. doi: 10.1142/S0219530515400059. Google Scholar

[2]

G. Albi, M. Burger, J. Haskovec, P. Markowich and M. Schlottbom, Continuum Modeling of Biological Network Formation, in Active Particles, vol. I. Modeling and Simulation in Science and Technology (eds. N. Bellomo, P. Degond and T. Tamdor), Birkhäuser, Boston, 2017, 1–48. Google Scholar

[3]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[4]

N. Dengler and J. Kang, Vascular patterning and leaf shape, Current Opinion in Plant Biology, 4 (2001), 50-56. doi: 10.1016/S1369-5266(00)00135-7. Google Scholar

[5]

A. EichmannF. Le NobleM. Autiero and P. Carmeliet, Guidance of vascular and neural network formation, Current Opinion in Neurobiology, 15 (2005), 108-115. doi: 10.1016/j.conb.2005.01.008. Google Scholar

[6]

A. Gmira and L. Véron, Large time behaviour of the solutions of a semilinear parabolic equation in $ \mathbb R^n$, Journal of Differential Equations, 53 (1984), 258-276. doi: 10.1016/0022-0396(84)90042-1. Google Scholar

[7]

J. HaskovecP. Markowich and B. Perthame, Mathematical analysis of a PDE system for biological network formation, Comm. Partial Differential Equations, 40 (2015), 918-956. doi: 10.1080/03605302.2014.968792. Google Scholar

[8]

J. HaskovecP. MarkowichB. Perthame and M. Schlottbomc, Notes on a PDE system for biological network formation, Nonlinear Analysis, 138 (2016), 127-155. doi: 10.1016/j.na.2015.12.018. Google Scholar

[9]

D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Physical Review Letters, 111 (2013), 138701. doi: 10.1103/PhysRevLett.111.138701. Google Scholar

[10]

D. Hu, Optimization, Adaptation, and Initialization of Biological Transport Networks, Notes from lecture, 2014.Google Scholar

[11]

E. Katifori, G. Szollosi and M. Magnasco, Damage and fluctuations induce loops in optimal transport networks, Physical Review Letters, 104 (2010), 048704. doi: 10.1103/PhysRevLett.104.048704. Google Scholar

[12]

H. Lin and Z. Xiang, Global well-posedness for 2D incompressible magneto-micropolar fluid system with partial viscosity, Sci. China Math., in press, (2019), http://engine.scichina.com/doi/10.1007/s11425-018-9427-6. doi: 10.1007/s11425-018-9427-6. Google Scholar

[13]

J. Liu and X. Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, Journal of Differential Equations, 264 (2018), 5489-5526. doi: 10.1016/j.jde.2018.01.001. Google Scholar

[14]

R. Malinowski, Understanding of leaf development-the science of complexity, Plants, 2 (2013), 396-415. doi: 10.3390/plants2030396. Google Scholar

[15]

O. Michel and J. Biondi, Morphogenesis of neural networks, Neural Processing Letters, 2 (1995), 9-12. doi: 10.1007/BF02312376. Google Scholar

[16]

Y. Peng and Z. Xiang, Global solutions to the coupled chemotaxis-fluids system in a 3D unbounded domain with finite depth, Math. Models Methods Appl. Sci., 28 (2018), 869-920. doi: 10.1142/S0218202518500239. Google Scholar

[17]

Y. Peng and Z. Xiang, Global existence and convergence rates to achemotaxis-fluids system with mixed boundary conditions, Journal of Differential Equations, 267 (2019), 1277-1321. doi: 10.1016/j.jde.2019.02.007. Google Scholar

[18]

X. RenJ. WuZ. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541. doi: 10.1016/j.jfa.2014.04.020. Google Scholar

[19]

X. RenZ. Xiang and Z. Zhang, Global well-posedness for the 2D MHD equations without magnetic diffusion in a strip domain, Nonlinearity, 29 (2016), 1257-1291. doi: 10.1088/0951-7715/29/4/1257. Google Scholar

[20]

X. RenZ. Xiang and Z. Zhang, Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion, Sci. China Math., 59 (2016), 1949-1974. doi: 10.1007/s11425-016-5145-2. Google Scholar

[21]

D. Sedmera, Function and form in the developing cardiovascular system, Cardiovascular Research, 91 (2011), 252-259. doi: 10.1093/cvr/cvr062. Google Scholar

[22]

J. Serrin, On the interior regularity of weak solutions of Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195. doi: 10.1007/BF00253344. Google Scholar

[23]

X. Xu, Life-span of smooth solutions to a PDE system with cubic nonlinearity, preprint, arXiv: 1706.06057.Google Scholar

[24]

X. Xu, Regularity theorems for a biological network formulation model in two space dimensions, Kinetic and Related Models, 11 (2018), 397-408. doi: 10.3934/krm.2018018. Google Scholar

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