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 (μ(x) ― yu) where μ belongs to a fixed subset of C0([a, b]). We prove that the knowledge of u at t = 0 and of u, ux at a single point x0 and for small times t E (0, ε) is sufficient to completely determine the couple (u(t, x), μ(x)) provided γ is known. Additionally, if uxx(t, x0) is also measured for t ∈ (0, s), the triplet (u(t, x), μ(x), γ) is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of u and ux at a single point x0 (and for t ∈ (0, ε)) are sufficient to obtain a good approximation of the coefficient μ(x). These numerical simulations also show that the measurement of the derivative ux is essential in order to accurately determine μ(x).