On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of the Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval (a, b) and we assume a nonlinear term of the form u (mu(x) - gamma u) where mu belongs to a fixed subset of C-0([a, b]). We prove that the knowledge of u at t = 0 and of u, u(x) at a single point x(0) and for small times t is an element of (0, epsilon) is sufficient to completely determine the couple (u(t, x), mu(x)) provided. is known. Additionally, if u(xx)(t, x(0)) is also measured for t is an element of (0, epsilon), the triplet (u(t, x), mu(x), gamma) is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of u and u(x) at a single point x(0) (and for t is an element of (0, epsilon)) are sufficient to obtain a good approximation of the coefficient mu(x). These numerical simulations also show that the measurement of the derivative u(x) is essential in order to accurately determine mu(x).