| Abstract: |
| The classical model for chemotaxis is the planar Keller-Segel system $$ u_t = \Delta u - \nabla\cdot ( u\nabla v ), \quad v(\cdot, t) = \frac 1{2\pi} \log 1{|\cdot |} * u(\cdot ,t) . $$ in $\R^2\times (0,\infty)$. A blow-up of finite mass solutions is expected to take place by aggregation, which is a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case $8\pi$, in which blow-up in infinite time takes place. We establish the stability of the phenomenon detected under arbitrary mass-preserving small perturbations and discuss new constructions in the finite time blow-up scenario. This is joint work with Juan Davila, Monica Musso, Federico Buseghin and Juncheng Wei. |
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