SMS scnews item created by Daniel Daners at Mon 27 Feb 2012 2127
Type: Seminar
Modified: Mon 27 Feb 2012 2132
Distribution: World
Expiry: 5 Mar 2012
Calendar1: 5 Mar 2012 1400-1500
CalLoc1: AGR Carslaw 829
Auth: daners@d220-237-40-101.mas801.nsw.optusnet.com.au (ddan2237) in SMS-WASM

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

### Kozono

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

## Abstract

In $$\mathbb R^n$$ ($$n \geq 3$$), 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 $$n \geq 4$$, 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 Cîrstea or Daniel Daners.

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