Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

TL;DR

This paper establishes bounds on the blow-up time for semi-discrete Keller-Segel models in 2D, demonstrating that these bounds match the continuous case and validating results through numerical simulations.

Contribution

It provides the first derivation of blow-up time bounds for various semi-discrete schemes in Keller-Segel models, extending continuous theory to discrete methods.

Findings

Bounds for blow-up time are consistent across multiple discretization schemes.

Numerical simulations confirm theoretical bounds and model behavior.

Discrete virial arguments effectively analyze semi-discrete chemotaxis models.

Abstract

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.