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

TL;DR

This paper studies a generalized aggregation-diffusion equation involving the p-Laplacian and an attraction kernel, identifying conditions for existence of solutions based on the balance between aggregation and diffusion effects.

Contribution

It establishes existence results for solutions in subcritical and critical regimes, highlighting the role of the parameters and initial conditions.

Findings

Existence of solutions in subcritical domain for all masses.

Existence of solutions in critical case for small initial mass.

Characterization of the competition between aggregation and diffusion.

Abstract

This work deals with the aggregation diffusion equation \[\partial_t \rho = \Delta_p\rho + \lambda div((K_a*\rho)\rho),\] 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.