| Abstract: |
| In this talk, we focus on the critical dynamics of the Keller-Segel model with $p$-Laplacian diffusion. We first establish a sharp threshold that distinguishes globally bounded solutions from finite-time blow-up. In the critical case, we rigorously construct backward self-similar blow-up solutions with compact support and radial monotonicity. These solutions exhibit concentration into a Dirac $\delta$-singularity at the blow-up time as their support shrinks to the origin. Additionally, we explore forward self-similar singular solutions, further characterizing their initial singularity. |
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