A critical parabolic estimate and application to nonlocal equations arising in chemotaxis

Abstract

We consider a nonlinear parabolic equation with exponential nonlinearity which contains a nonlocal multiplication factor. This equation models an aggregation of cells via interaction with a chemical substance. We concentrate on two dimensional geometry, where a critical population number exists beyond which there are no regular steady states. The analysis involves a comparison theorem for linear parabolic equations with L¹ sources, as well as a critical Sobolev inequality

Keywords:

• parabolic equations
• chemotaxis
• blow up

¹Tel. [phone] ; fax: [phone] ; e.mail:[email protected]

¹Tel. [phone] ; fax: [phone] ; e.mail:[email protected]

Notes

¹Tel. [phone] ; fax: [phone] ; e.mail:[email protected]