  Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing
function and $u_0$ be a possibly singular nonnegative initial function. We are
concerned with existence and nonexistence of a local in time nonnegative
solution in a uniformly local Lebesgue space of a semilinear heat equation \[
\begin{cases} \partial_tu=\Delta u+f(u) & \textrm{in}\
\mathbb{R}^N\times(0,T),\\ u(x,0)=u_0(x) & \textrm{in}\ \mathbb{R}^N
\end{cases} \] under mild assumptions on $f$. A relationship between a growth
of $f$ and an integrability of $u_0$ is studied in detail. Our existence
theorem gives a sharp integrability condition on $u_0$ in a critical and
subcritical cases, and it can be applied to a regularly or rapidly varying
function $f$. In a doubly critical case existence and nonexistence of a
nonnegative solution can be determined by special treatment. When
$f(u)=u^{1+2/N}[\log(u+e)]^{\beta}$, a complete classification of existence and
nonexistence of a nonnegative solution is obtained. We also show that the same
characterization as in Laister et. al. [11] is still valid in the closure of
the space of bounded uniformly continuous functions in the space $L^r_{\rm
ul}(\mathbb{R}^N)$. Main technical tools are a monotone iterative method,
$L^p$-$L^q$ estimates, Jensen's inequality and differential inequalities.
