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

This work deals with the aggregation diffusion equation $${\mathrm{\partial }}_t\rho ={\mathrm{\Delta }}_p\rho +\lambda div((K_a\ast \rho )\rho ),$$ where $K_a(x)=\frac{x}{\mathrm{\mid }x{\mathrm{\mid }}^a}$ is an attraction kernel and ${\mathrm{\Delta }}_p$ 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.