  We study the Cauchy problem for the semilinear fractional heat equation
$u_{t}=\triangle^{\alpha/2}u+f(u)$ with non-negative initial value $u_{0}\in
L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For
$f$ satisfying the Osgood-type condition
$\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial
conditions such that the equation has no local solution in
$L^{1}_{loc}(\mathbb{R}^{n})$.
