| Abstract: |
| 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 under additional moment assumptions on the initial data. A common approach in the literature is to impose such conditions, as they provide effective control of the behavior of solutions at spatial infinity. However, these assumptions are extrinsic to the intrinsic scaling structure underlying the critical mass phenomenon. In the absence of moment conditions, classical energy-based methods cease to be effective, and the problem becomes considerably more delicate at the critical threshold due to the degeneracy of the underlying energy structure. In this talk, we establish global-in-time existence for initial data at the critical mass without imposing any additional assumptions. The proof relies on a reconstructed Lyapunov functional and refined dissipative estimates that recover sufficient control of the dynamics in the whole space. |
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