On the Instability of Threshold Solutions of Reaction-Diffusion Equations, and Applications to Optimization Problems

Abstract. The aim of this paper is to address optimization problems in bistable reaction-diffusion equations, motivated by population dynamics questions. Namely, we want to characterize the initial data guaranteeing the persistence of the population at large times, and at the same time minimizing a cost function. This leads us to first derive a technical result. Namely, consider an initial datum leading to convergence to an intermediate threshold solution belonging to [Formula: see text]. We characterize the perturbations of the initial datum such that the associated solution is strictly larger or smaller than the threshold solution at large times. This condition involves an adjoint solution, which satisfies a backward parabolic equation and is uniquely defined [J. Húska and P. Poláčik, Discrete Contin. Dyn. Syst., 20 (2008), pp. 81–113]. We then provide two applications of our result. We first address an open problem stated in [J. Garnier, L. Roques, and F. Hamel, Bull. Math. Biol., 74 (2012), pp. 453–473] in dimension 1. Namely, we compute the derivative of the critical length [Formula: see text] associated with the initial datum [Formula: see text], that is the length [Formula: see text] above (resp., below) which the associated solution converges to 1 (resp., 0) at large times. Lastly, again in dimension 1, we prove the existence and characterize with a bathtub principle the initial datum minimizing some cost function and guaranteeing at the same time the persistence of the solution at large times.