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 \(\mathbb R^n\) (\(n \geq 3\)), 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.

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