  In this paper, we consider the fractional heat equation
$u_{t}=\triangle^{\alpha/2}u+f(u)$ with Dirichlet boundary conditions on the
ball $B_{R}\subset \mathbb{R}^{d}$, where $\triangle^{\alpha/2}$ is the
fractional Laplacian, $f:[0,\infty)\rightarrow [0,\infty)$ is continuous and
non-decreasing. We present the characterisations of $f$ to ensure the equation
has a local solution in $L^{q}(B_{R})$ provided that the non-negative initial
data $u_{0}\in L^{q}(B_{R})$. For $q>1$ and $1<\alpha\leq 2$, we show that the
equation has a local solution in $L^{q}(B_{R})$ if and only if
$\lim_{s\rightarrow \infty}\sup s^{-(1+\alpha q/d)}f(s)=\infty$; and for $q=1$
and $1<\alpha\leq 2$ if and only if
$\int_{1}^{\infty}s^{-(1+\alpha/d)}F(s)ds<\infty$, where $F(s)=\sup_{1\leq
t\leq s}f(t)/t$. When $\lim_{s\rightarrow 0}f(s)/s<\infty$, the same
characterisations holds for the fractional heat equation on the whole space
$\mathbb{R}^{d}$.
