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

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 $t\in (0,\epsilon )$ 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 $t\in (0,\epsilon )$, 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 $t\in (0,\epsilon )$) 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 $\mu (x)$.

References in corpus (1)

• Inverse problems for two by two reaction-diffusion system using a Carleman estimate with one observation

Cited by in corpus (1)

• Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions