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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[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.

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