| 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 occur by aggregation, a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case $8\pi$, where blow-up takes place in infinite time. We establish the stability of the phenomenon detected under arbitrary mass-preserving small perturbations and present new constructions in the finite time blow-up scenario. The results presented are in collaboration with Federico Buseghin, Juan Davila, Jean Dolbeault, Monica Musso, and Juncheng Wei. |
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