分野別セミナー

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.