Stable Singularity Formation for the Keller-Segel System in Three Dimensions.

Area of Science:

• Mathematical Physics
• Partial Differential Equations
• Nonlinear Dynamics

Background:

• The parabolic-elliptic Keller-Segel system models chemotaxis and exhibits complex dynamics, including singularity formation.
• Self-similar solutions are crucial for understanding blowup behavior in this system, particularly in the mass supercritical regime.

Purpose of the Study:

• To rigorously prove the conjecture of nonlinear radial stability for self-similar blowup solutions of the Keller-Segel system in dimensions $$.
• To establish a general and robust analytical framework for studying parabolic models.

Main Methods:

• Reformulation of the Keller-Segel system in similarity variables.
• Analysis of Cauchy evolution in intersection Sobolev spaces using semigroup theory.
• Application of a specialized spectral problem technique developed by Glogić and Schörkhuber (2020).

Main Results:

• The conjecture on the nonlinear radial stability of self-similar blowup solutions for the Keller-Segel system in $$ is proven.
• This represents the first established result on stable self-similar blowup for the Keller-Segel system.
• The methodology is readily extendable to higher dimensions and other parabolic models.

Conclusions:

• The study confirms the stability of a key class of solutions for the Keller-Segel system.
• The developed analytical approach is versatile, paving the way for broader applications in studying parabolic systems.
• This work significantly advances the understanding of singularity formation and stability in chemotaxis models.