| Abstract: |
| The talk will be concerned with aggregation diffusion equations and in particular with the well-known Keller-Segel model for chemotaxis, where a particle population is subject to diffusion and to advection in the direction of the gradient of the concentration of a substance produced by the particles themselves. Such a concentration is defined at each point in terms of a non-local integral involving the whole distribution of particles. I will briefly recall the main facts about the Keller-Segel system with particular attention to the critical cases, in terms of diffusion exponent and mass and depending on the dimension, and then present some recent results obtained in collaboration with Elbar and Fernandez-Jimenez. In our recent work we obtained a new estimate on the Laplacian of the pressure associated with the solution, which allows one to obtain or recover global existence results, regularity and regularization, and information of the asymptotic behavior and decay. |
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