p-Laplacian Keller-Segel Equation: Fair Competition and Diffusion Dominated Cases

Abstract

This work deals with the aggregation diffusion equation \[\partial_t ρ= Δ_pρ+ λdiv((K_a*ρ)ρ),\] where $K_a(x)=\frac{x}{|x{|}^a}$ is an attraction kernel and ${\Delta }_p$ is the so called $p$-Laplacian. We show that the domain $a<p(d+1)-2d$ is subcritical with respect to the competition between the aggregation and diffusion by proving that there is existence unconditionally with respect to the mass. In the critical case we show existence of solution in a small mass regime for an $L\ln L$ initial condition.

7 pages, 1 figure