| 概要 |
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.
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