Some inverse stability results for the bistable reaction–diffusion equation using Carleman inequalities
Muriel Boulakia, Céline Grandmont, Axel Osses · Comptes Rendus Mathématique · 2009
We consider the bistable equation v t − Δ v = f ( v , x ) , f ( v , x ) = a ( x ) v ( 1 − v ) ( v − α ( x ) ) with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 with regular boundary. For this equation, we prove Lipschitz stability for the inverse problem of recovering parameters a and α from measurements of v in ( 0 , T ) × ω , where ω is an arbitrary nonempty open subset of Ω and measurements of v ( t 0 ) in the whole domain Ω at some positive time t 0 such that 0 < t 0 < T . The result is based in some suitable global Carleman estimate for the nonlinear problem.