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

TL;DR

This paper investigates the uniqueness of identifying nonlinear reaction terms and coefficients in a reaction-diffusion equation from localized measurements, combining theoretical proofs with numerical simulations to demonstrate practical applicability.

Contribution

It establishes conditions under which the nonlinear term and coefficients can be uniquely determined from minimal localized measurements in a reaction-diffusion model.

Findings

Single-point measurements of $u$ and $u_x$ suffice for coefficient identification.

Measuring $u_{xx}$ enables full determination of all parameters including $eta$.

Numerical simulations confirm the practical effectiveness of the proposed measurement strategy.

Abstract

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of 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 $t\in (0,\varepsilon)$ is sufficient to completely determine the couple $(u(t,x),\mu(x))$ provided $\gamma$ is known. Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$, 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 $t\in (0,\varepsilon)$) 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 $\mu(x)$.