Existence and uniqueness theorem of weak solutions to the parabolic-elliptic Keller-Segel system

Hideo Kozono
Tohoku University, Japan
5th March 2012, 2-3pm, Carslaw 829 (Access Grid Room)

Abstract

In ${\mathbb{R}}^n$ ($n\ge 3$), we first define a notion of weak solutions to the Keller-Segel system of parabolic-elliptic type in the scaling invariant class $L^s((0,T);L^r({\mathbb{R}}^n))$ for $2/s+n/r=2$ with $n/2<r<n$. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in $L^{n/2}({\mathbb{R}}^n)$. We prove also their uniqueness. As for the marginal case when $r=n/2$, we show that if $n\ge 4$, then the class $C([0,T);L^{n/2}({\mathbb{R}}^n))$ enables us to obtain the only weak solution.

Check also the PDE Seminar page. Enquiries to Florica C�rstea or Daniel Daners.