Blowup criterion of classical solutions for a parabolic-elliptic system in space dimension 3

Li, Bin; Li, Yuxiang

doi:10.1090/proc/15637

Blowup criterion of classical solutions for a parabolic-elliptic system in space dimension 3
HTML articles powered by AMS MathViewer

by Bin Li and Yuxiang Li

Proc. Amer. Math. Soc. 149 (2021), 5291-5303

DOI: https://doi.org/10.1090/proc/15637

Published electronically: September 24, 2021

PDF | Request permission

Abstract:

This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.

References

Giacomo Albi, Marco Artina, Massimo Foransier, and Peter A. Markowich, Biological transportation networks: modeling and simulation, Anal. Appl. (Singap.) 14 (2016), no. 1, 185–206. MR 3438650, DOI 10.1142/S0219530515400059
Giacomo Albi, Martin Burger, Jan Haskovec, Peter Markowich, and Matthias Schlottbom, Continuum modeling of biological network formation, Active particles. Vol. 1. Advances in theory, models, and applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017, pp. 1–48. MR 3644587
Martin Burger, Jan Haskovec, Peter Markowich, and Helene Ranetbauer, A mesoscopic model of biological transportation networks, Commun. Math. Sci. 17 (2019), no. 5, 1213–1234. MR 4044188, DOI 10.4310/CMS.2019.v17.n5.a3
Jan Haskovec, Henrik Jönsson, Lisa Maria Kreusser, and Peter Markowich, Auxin transport model for leaf venation, Proc. A. 475 (2019), no. 2231, 20190015, 23. MR 4044080, DOI 10.1098/rspa.2019.0015
Jan Haskovec, Lisa Maria Kreusser, and Peter Markowich, ODE- and PDE-based modeling of biological transportation networks, Commun. Math. Sci. 17 (2019), no. 5, 1235–1256. MR 4044189, DOI 10.4310/CMS.2019.v17.n5.a4
Jan Haskovec, Lisa Maria Kreusser, and Peter Markowich, Rigorous continuum limit for the discrete network formation problem, Comm. Partial Differential Equations 44 (2019), no. 11, 1159–1185. MR 3995094, DOI 10.1080/03605302.2019.1612909
Jan Haskovec, Peter Markowich, and Benoit Perthame, Mathematical analysis of a PDE system for biological network formation, Comm. Partial Differential Equations 40 (2015), no. 5, 918–956. MR 3306822, DOI 10.1080/03605302.2014.968792
Jan Haskovec, Peter Markowich, Benoît Perthame, and Matthias Schlottbom, Notes on a PDE system for biological network formation, Nonlinear Anal. 138 (2016), 127–155. MR 3485142, DOI 10.1016/j.na.2015.12.018
D. Hu and D. Cai, Adaptation and optimization of biological transport networks, Phys. Rev. Lett. 111 (2013), no 13, 138701, 4 pp.
Dan Hu and David Cai, An optimization principle for initiation and adaptation of biological transport networks, Commun. Math. Sci. 17 (2019), no. 5, 1427–1436. MR 4044197, DOI 10.4310/CMS.2019.v17.n5.a12
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822, DOI 10.1090/mmono/023
Bin Li, Long time behavior of the solution to a parabolic-elliptic system, Comput. Math. Appl. 78 (2019), no. 10, 3345–3362. MR 4018641, DOI 10.1016/j.camwa.2019.05.005
Bin Li, On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks, Kinet. Relat. Models 12 (2019), no. 5, 1131–1162. MR 4027080, DOI 10.3934/krm.2019043
Bin Li, Global existence and decay estimates of solutions of a parabolic-elliptic-parabolic system for ion transport networks, Results Math. 75 (2020), no. 2, Paper No. 45, 28. MR 4075299, DOI 10.1007/s00025-020-1172-y
Bin Li and Jieqiong Shen, Classical solution of a PDE system stemming from auxin transport model for leaf venation, Proc. Amer. Math. Soc. 148 (2020), no. 6, 2565–2578. MR 4080897, DOI 10.1090/proc/14951
Jian-Guo Liu and Xiangsheng Xu, Partial regularity of weak solutions to a PDE system with cubic nonlinearity, J. Differential Equations 264 (2018), no. 8, 5489–5526. MR 3760183, DOI 10.1016/j.jde.2018.01.001
James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187–195. MR 136885, DOI 10.1007/BF00253344
Xiangsheng Xu, Regularity theorems for a biological network formulation model in two space dimensions, Kinet. Relat. Models 11 (2018), no. 2, 397–408. MR 3810832, DOI 10.3934/krm.2018018
X. Xu, Partial regularity of weak solutions and life-span of smooth solutions to a PDE system with cubic nonlinearity, SN Partial Differ. Equ. Appl. 1 (2020), no 4, 18, 31 pp.
Xiangsheng Xu, Global existence of strong solutions to a biological network formulation model in $2+1$ dimensions, Discrete Contin. Dyn. Syst. 40 (2020), no. 11, 6289–6307. MR 4147351, DOI 10.3934/dcds.2020280