Stable estimation of two coefficients in a nonlinear Fisher-KPP equation
2013
Cristofol, Michel | Roques, Lionel
We consider the inverse problem of determining two non-constant coe cients in a nonlinear parabolic equation of the Ficher-Kolmogorov-Petrovsky- Piskunov type. For the equation u(t)=D delta u+ µ(x)u - gamma (x) u² in (0;T) * omega (lettre majuscule grecque), which corresponds to a classical model of population dynamics in a bounded heterogeneous environment, our results give a stability inequality between the couple of coe cients ( µ;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 [ω CC(sous ensemble de) oméga (lettre majuscule grecque)] during a nite time interval; and on the boundary of omega (lettre majuscule grecque)at all times t appartient à (0;T). The proof relies on parabolic estimates together with parabolic maximum principle and Hopf's lemma which enable us to use a Carleman inequality. This work extends previous studies on stable determination of non-constant coe cients in parabolic equations, as it deals with two coe cients and with a nonlinear term. A consequence of our results is the uniqueness of the couple of coe cients (µ ;gamma); given the observation of u. This uniqueness result was obtained in a previous article but in the one-dimensional case only.
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