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Journal Articles Nonlinearity Year : 2010

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

Lionel Roques

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Michel Cristofol

#### 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 $\mu(x)$.

### Dates and versions

hal-00488933 , version 1 (03-06-2010)

### Identifiers

• HAL Id : hal-00488933 , version 1
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### Cite

Lionel Roques, Michel Cristofol. On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation. Nonlinearity, 2010, 23 (3), pp.675-686. ⟨10.1088/0951-7715/23/3/014⟩. ⟨hal-00488933⟩

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