Contents Online
Communications in Mathematical Sciences
Volume 20 (2022)
Number 7
Blow-up time of strong solutions to a biological network formation model in high space dimensions
Pages: 2029 – 2052
DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n7.a10
Author
Abstract
We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by [D. Cai and D. Hu, Phys. Rev. Lett., 111:138701, 2013]. The model is represented by an initial and boundary value problem for an elliptic-parabolic system with cubic nonlinearity. We obtain an algebraic equation for the possible blow-up time of strong solutions. The equation yields information on how various given data may contribute to the blow-up of solutions. As a by-product of our development, we establish a $W^{1,q}$ estimate for solutions to an elliptic equation which shows the explicit dependence of the upper bound on the elliptic coefficients.
Keywords
biological network formation, blow-up time, existence
2010 Mathematics Subject Classification
35A01, 35B44, 35B65, 35D35, 35Q92
Received 27 September 2021
Received revised 3 February 2022
Accepted 12 February 2022
Published 21 October 2022