Stable determination of coefficients in semilinear parabolic system with dynamic boundary conditions

Abstract

In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.

Determining a parabolic system by boundary observation of its non-negative solutions with applications

“…and consider the new system associated to (u (1) , v (1) ). For F, G ∈ C 1 , we have (u(x, t; 0), v(x, t; 0)) = (0, 0), so ∂ t u (1) (x, t) − µ∆u (1) (x, t) = 0 and ∂ t v (1) (x, t) − ν∆v (1) (x, t) = β 10 (x, t)u (1) (x, t) + β 01 (x, t)v (1) (x, t).…”

“…Then, u (1) satisfies the well-posed heat equation such that u (1) ∈ C 2+α,1+α/2 ( Q), which is strictly positive by the maximum principle. Extending f 1 (x, t) to Ω which we still denote by f 1 (x, t), ū := u (1) − f 1 (x, t) satisfies the well-posed heat equation…”

“…There has been much interest in inverse problems for coupled parabolic systems, and we refer to [15,8,2,5,1] and the references therein for examples in the literature of various such works. However, most of these works aim to recover the conductivity of the system, which is related to the self-diffusion coefficients of Section 1.2 when the PDE is written in non-divergence form.…”

“…Then, by direct calculations, we have the second order linearisation (1) ] 2 + β 10 u (2) + β 01 v (2) in Q, u (2)…”

“…We next show the result for m = 2, by considering the third order linearisation. Defining u (3) , v (3) inductively, we have the third order linearisation: 3) − µ∆u (3) = 6α 30 [u (1) ] 3 + 6α 03 [v (1) ] 3 + 6α 12 u (1) [v (1) ] 2 + 6α 21 [u (1) ] 2 v (1) in Q, ∂ t v (3) − ν∆v (3) = 6β 30 [u (1) ] 3 + 6β 03 [v (1) ] 3 + 6β 12 u (1) [v (1) ] 2 + 6β 21 [u (1) ] 2 v (1) +6β 20 u (1) u (2)…”