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



Xiangsheng Xu (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Miss., U.S.A.)


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.


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