<p>We study the Cauchy problem for the two dimensional fully parabolic Keller–Segel system at the critical mass. Global-in-time existence at the critical mass is known under radial symmetry or additional moment assumptions on the initial data. However, such weighted conditions are extrinsic to the intrinsic scaling structure underlying the critical mass phenomenon. In the absence of moment assumptions, classical energy-based approaches cease to be effective, and the problem becomes substantially more delicate at the critical threshold due to the degeneracy of the underlying energy structure. In this paper, we establish global-in-time existence for general initial data at the critical mass without imposing any symmetry or moment conditions. The proof relies on a reconstructed Lyapunov functional and refined dissipative estimates that recover sufficient control of the dynamics in the whole space setting.</p>

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Global existence for the fully parabolic Keller–Segel system with critical mass on the plane

  • Tatsuya Hosono

摘要

We study the Cauchy problem for the two dimensional fully parabolic Keller–Segel system at the critical mass. Global-in-time existence at the critical mass is known under radial symmetry or additional moment assumptions on the initial data. However, such weighted conditions are extrinsic to the intrinsic scaling structure underlying the critical mass phenomenon. In the absence of moment assumptions, classical energy-based approaches cease to be effective, and the problem becomes substantially more delicate at the critical threshold due to the degeneracy of the underlying energy structure. In this paper, we establish global-in-time existence for general initial data at the critical mass without imposing any symmetry or moment conditions. The proof relies on a reconstructed Lyapunov functional and refined dissipative estimates that recover sufficient control of the dynamics in the whole space setting.