On a class of diffusion-aggregation equations

Yuming Paul Zhang

doi:10.3934/dcds.2020066

Article Contents

2020, Volume 40, Issue 2: 907-932. Doi: 10.3934/dcds.2020066

This issue Previous Article Dimensions of $ C^1- $average conformal hyperbolic sets Next Article Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

On a class of diffusion-aggregation equations

Yuming Paul Zhang^,

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90025, USA

^* Corresponding author: Yuming Paul Zhang
^* Corresponding author: Yuming Paul Zhang

Received: March 2019

Revised: August 2019

Published: November 2019

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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}) $. The equation is related to biological aggregation models. We analyze the regime where the diffusive force is stronger than the aggregation effect. In such regime, we show the existence and uniform boundedness of solutions in the case either $ s>\frac{1}{2} $ or $ m<2 $. Hölder regularity is obtained when $ d\geq3, s>1/2 $ and uniqueness is proved when $ s\geq 1 $.

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Mathematics Subject Classification: Primary: 35K55, 45G05; Secondary: 35K65, 35B65.

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