Blow-up time of strong solutions to a biological network formation model in high space dimensions

TL;DR

This paper analyzes conditions under which strong solutions to a biological network formation model may blow up in high-dimensional spaces, providing algebraic criteria and estimates related to solution behavior.

Contribution

It introduces an algebraic equation for blow-up time and derives a $W^{1,q}$ estimate showing how data influences solution blow-up in a biological network model.

Findings

Derived an algebraic equation for blow-up time.

Established a $W^{1,q}$ estimate with explicit data dependence.

Identified data contributions to solution blow-up.

Abstract

We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic system with cubic non linearity. 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.