Stability of Diffusion Coefficients in an Inverse Problem for the Lotka-Volterra Competition System

Abstract

First we establish a Carleman estimate for Lotka-Volterra competition-diffusion system of three equations with variable coefficients. Then the internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of retrieving two smooth diffusion coefficients in the given parabolic system for the dimension n ≤ 3. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic system.

Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology

“…The paper by Cristofol et al [11] obtains the stability results for reaction-diffusion system of two equations with constant coefficients using a Carleman estimate. Then Sakthivel et al [21] established the stability results for Lotka-Volterra competitiondiffusion system of three equations with variable diffusion coefficients. Our inverse stability results are new because system (6) contains a strong coupling term.…”

“…Our inverse stability results are new because system (6) contains a strong coupling term. The technics we shall discuss are similar to the framework using Carleman estimates for inverse problems but the obtained estimates differs from those of [24], [21] because of the strongly coupled terms.…”

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

“…For example, Cristofol et al [29] discuss the simultaneous reconstruction of one coefficient and initial conditions for the reaction-diffusion system from the measurement of one solution over (t 0 , T ) × Ω and some measurement at fixed timeT ∈ (t 0 , T ), and Benabdallah et al [30] studies the inverse problems of determining all (or some of) the coefficients from the observations on an arbitrary subdomain over a time interval of only one component and data for two components at a fixed positive time θ over the whole spatial domain. The paper by Baranibalan et al [31] studies the inverse problem of determining a diffusion coefficient for a phase field system of two equations from one observation and Sakthivel et al [32] successfully apply a similar method for determining two diffusion coefficients of a Lotka-Volterra competitive diffusion system of three equations with two observations. Our interest in this paper is in determining the coefficients a(x) and c(x) from lateral data for only one component, which can be briefly described as follows.…”

Ionic parameters identification of an inverse problem of strongly coupled PDE’s system in cardiac electrophysiology using Carleman estimates

“…Our inverse stability results are new because the bidomain system contains a strong coupling term. The technics we shall discuss are similar to the framework using Carleman estimates for inverse problems but the obtained estimates differs from those of [17,44,53] because of the strongly coupled terms. We use the same strategy introduced by Wu and Yu in [51] to convert the strongly coupled terms to the gradient of the extra-cellular potential u e .…”