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In this article, we propose in-depth analysis and characterisation of theoptimisers of the following optimisation problem: how to choose the initialcondition $u_0$ in order to maximise the spatial integral at a given time ofthe 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 togive a characterisation of the behaviour of the optimiser $\overline{u}_0$ whenit does not saturate the $L^\infty$ constraints, which is a key step inimplementing efficient numerical algorithms. We give such a characterisationunder 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 usingtwo-scale asymptotic expansions. We then show how well-known isoperimetricinequalities yield a full characterisation of maximisers when $f$ is convex.Finally, we provide several numerical simulations in one and two dimensionsthat illustrate and exemplify the fact that such characterisationssignificantly improves the computational time. All our theoretical results arein the one-dimensional case and we offer several comments about possiblegeneralisations to other contexts, or obstructions that may prohibit doing so.

nonlinearity

Optimisation of the total population size with respect to the initial  condition for semilinear parabolic equations: Two-scale expansions and  symmetrisations