On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \R^N$, where the initial datum $u(0,x)=u_0(x)$ is such that $\lim_{t\to +\infty} u(t,x)=W(x)$ for all $x$, with $W$ a steady state in $H^1(\R^N)$. We characterize the perturbations $h$ such that, if $u^h$ is the solution associated with the initial datum $u_0+h$, then, if $h$ is small enough in a sense, one has $u^h(t,x)>W(x)$ (resp. $u(t,x)0$ for all $x$