On a Class of Diffusion-Aggregation Equations

Abstract

We investigate the diffusion-aggregation equations with degenerate diffusion $\Delta u^m$ and singular interaction kernel $\mathcal{K}_s = (-\Delta)^{-s}$ with $s\in(0,\frac{d}{2})$. We analyze the regime %($m>2-2s/d$, $d$ is the dimension) where the diffusive forces are stronger than the aggregation forces. In such regime, we show existence, uniform boundedness and H\"{o}lder regularity of solutions in the case that either $s>\frac{1}{2}$ or $m<2$. Uniqueness is proved for kernels with $s>1$.