A Nonlocal Semilinear Neumann Problem and Applications to a Chemotaxis System

Matteo Talluri

September 30th, 2025. 

Time: 5 PM 

Where: Seminario 1 (Dip. Mat. UNIBO)

Abstract: In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when the steady states of the Keller–Segel model are considered with nonlocal chemical concentration diffusion. There are different possible notions of "non-local Neumann conditions" in the fractional setting. We will study the system both with spectral conditions, as was done by Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci) as was done by Cinti and Colasuonno. While for the spectral boundary condition there exists an extension theorem analogous to that of Caffarelli and Silvestre, such a theorem does not exist for integral boundary conditions. This makes it more difficult to obtain Liouville-type theorems and to show that the system has non-trivial positive solutions if the concentration parameter is big enough. Based on a joint project with Eleonora Cinti.