Oscillation Decay and Regularity in Fast-Diffusion Chemotaxis

In chemotaxis systems with nonlinear diffusion, the interplay between diffusion and chemically directed movement gives rise to challenging questions concerning the regularity of weak solutions. In this talk, we study a parabolic-elliptic Keller-Segel system of fast-diffusion type, with a source term depending on both the cell density and its gradient. Using an intrinsic De Giorgi-DiBenedetto approach, we show that locally bounded weak solutions are locally Hölder continuous. Finally, we briey discuss some open problems and an ongoing research direction concerning the local regularity of the spatial gradient of solutions. This talk is based on joint work with Monica Marras and Stella Vernier-Piro.