  We consider the scalar semilinear heat equation $u_t-\Delta u=f(u)$, where
$f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing but need not
be convex. We completely characterise those functions $f$ for which the
equation has a local solution bounded in $L^q(\Omega)$ for all non-negative
initial data $u_0\in L^q(\Omega)$, when $\Omega\subset{\mathbb R}^d$ is a
bounded domain with Dirichlet boundary conditions. For $q\in(1,\infty)$ this
holds if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and for
$q=1$ if and only if $\int_1^\infty s^{-(1+2/d)}F(s)\,{\rm d}s<\infty$, where
$F(s)=\sup_{1\le t\le s}f(t)/t$. This shows for the first time that the model
nonlinearity $f(u)=u^{1+2q/d}$ is truly the `boundary case' when
$q\in(1,\infty)$, but that this is not true for $q=1$.
  The same characterisation results hold for the equation posed on the whole
space ${\mathbb R}^d$ provided that in addition $\limsup_{s\to0}f(s)/s<\infty$.
