Stable estimation of two coefficients in a nonlinear Fisher-KPP equation

We consider the inverse problem of determining two non-constant coefficients in a nonlinear parabolic equation of the Fisher-Kolmogorov-Petrovsky-Piskunov type. For the equation u(t) = D Delta u + mu(x) u - gamma (x)u(2) in (0, T) x Omega, which corresponds to a classical model of population dynamics in a bounded heterogeneous environment, our results give a stability inequality between the couple of coefficients (mu, gamma) and some observations of the solution u. These observations consist in measurements of u: in the whole domain Omega at two fixed times, in a subset omega subset of subset of Omega during a finite time interval and on the boundary of Omega at all times t is an element of (0, T). The proof relies on parabolic estimates together with the parabolic maximum principle and Hopf's lemma which enable us to use a Carleman inequality. This work extends previous studies on the stable determination of non-constant coefficients in parabolic equations, as it deals with two coefficients and with a nonlinear term. A consequence of our results is the uniqueness of the couple of coefficients (mu, gamma), given the observation of u. This uniqueness result was obtained in a previous paper but in the one-dimensional case only.