P-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases

Abstract

This work deals with the aggregation diffusion equation∂tρ=Δpρ+λdiv((Ka⁎ρ)ρ), where Ka(x)=x|x|a is an attraction kernel and Δp 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 the existence of a solution unconditionally with respect to the mass. In the critical case, we show the existence of a solution in a small mass regime for an Lln⁡L initial condition.