  In this article, we propose in-depth analysis and characterisation of the
optimisers of the following optimisation problem: how to choose the initial
condition $u_0$ in order to maximise the spatial integral at a given time of
the solution of the semilinear equation $u_t-\Delta u=f(u)$, under $L^\infty$
and $L^1$ constraints on $u_0$? Our contribution in the present paper is to
give a characterisation of the behaviour of the optimiser $\overline{u}_0$ when
it does not saturate the $L^\infty$ constraints, which is a key step in
implementing efficient numerical algorithms. We give such a characterisation
under mild regularity assumptions by proving that in that case $\overline{u}_0$
can only take values in the "zone of concavity" of $f$. This is done using
two-scale asymptotic expansions. We then show how well-known isoperimetric
inequalities yield a full characterisation of maximisers when $f$ is convex.
Finally, we provide several numerical simulations in one and two dimensions
that illustrate and exemplify the fact that such characterisations
significantly improves the computational time. All our theoretical results are
in the one-dimensional case and we offer several comments about possible
generalisations to other contexts, or obstructions that may prohibit doing so.
