For the 2D Keller-Segel equation at critical mass 8π with finite second momentum, all solutions converge asymptotically to a renormalized stationary state concentrating at the center of mass on a logarithmic-in-time scale, without symmetry assumptions.
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Radial solutions to the 2D parabolic-elliptic Keller-Segel system exhibit subcritical convergence to a self-similar expander, critical infinite-time concentration at logarithmic rate, or supercritical type-II blow-up, for all radial data with finite second momentum.
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Determination of the long-time dynamics for the 2D Keller-Segel equation at critical mass
For the 2D Keller-Segel equation at critical mass 8π with finite second momentum, all solutions converge asymptotically to a renormalized stationary state concentrating at the center of mass on a logarithmic-in-time scale, without symmetry assumptions.
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Classification of the dynamics of radial solutions to the 2D parabolic-elliptic Keller-Segel System
Radial solutions to the 2D parabolic-elliptic Keller-Segel system exhibit subcritical convergence to a self-similar expander, critical infinite-time concentration at logarithmic rate, or supercritical type-II blow-up, for all radial data with finite second momentum.