Existence of Finite Time Blow-Up in Keller-Segel System

Area of Science:

• Mathematical biology
• Partial differential equations
• Nonlinear dynamics

Background:

• The Keller-Segel system is a classical model for chemotaxis, describing the movement of organisms in response to chemical signals.
• Understanding the long-term behavior of solutions, particularly blow-up phenomena, is crucial for comprehending biological pattern formation.

Purpose of the Study:

• To investigate the existence of finite-time blow-up solutions for the Keller-Segel system in 2D.
• To characterize the asymptotic profile of these blow-up solutions.
• To generalize existing results on stable blow-up dynamics.

Main Methods:

• Construction of specific initial data for the Keller-Segel system.
• Analysis of the system's behavior near the blow-up time.
• Asymptotic analysis to determine the blow-up profile.

Main Results:

• Existence of solutions that blow up in finite time for a range of initial total mass (m).
• The blow-up profile is characterized by a sum of localized, self-similar structures.
• The scaling of the blow-up radius (λ_j(t)) is described, involving logarithmic terms and the Euler-Mascheroni constant.

Conclusions:

• The study provides a generalized construction for blow-up solutions in the Keller-Segel model.
• The findings extend the understanding of pattern formation and singularities in chemotaxis models.
• This work contributes to the mathematical theory of reaction-diffusion systems with aggregation.