On a fractional semilinear Neumann problem arising in Chemotaxis

TL;DR

This paper investigates a fractional nonlocal Neumann problem related to chemotaxis, establishing existence of non-constant solutions for small diffusion parameters and proving solutions are constant for large parameters.

Contribution

It extends previous results to any fractional power of the Laplacian, providing new existence and qualitative properties of solutions in the chemotaxis model.

Findings

Existence of non-constant solutions for small diffusion parameter .

Solutions are necessarily constant for large .

Extension of results to all fractional powers s  (0,1).

Abstract

We study a semilinear and nonlocal Neumann problem, which is the fractional analogue of the problem considered by Lin--Ni--Takagi in the '80s. The model under consideration arises in the description of stationary configurations of the Keller--Segel model for chemotaxis, when a nonlocal diffusion for the concentration of the chemical is considered. In particular, we extend to any fractional power $s\in (0,1)$ of the Laplacian (with homogeneous Neumann boundary conditions) the results obtained in [20] for $s=1/2$. We prove existence and some qualitative properties of non--constant solutions when the diffusion parameter $\varepsilon$ is small enough, and on the other hand, we show that for $\varepsilon$ large enough any solution must be necessarily constant.