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

In this article, we propose in-depth analysis and characterisation of the optimisers of the following optimisation problem: how to choose the initial condition u₀ in order to maximise the spatial integral at a given time of the solution of the semilinear equation u_t −Δu = f(u), under L^∞ and L¹ constraints on u₀? 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^∞ 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 improve 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.