Generic well-posedness of a linear inverse parabolic problem with diffusion parameters

Abstract

We study a linear inverse parabolic problem with final overdetermination, which is parametrized by diffusion constants. We show that this inverse parabolic problem is well-posed for values of diffusion parameters except for a finite set of values. The keys are the analytic dependence on parameters and the analytic Fredholm alternative.

A novel method to solve inverse source problem for advection-diffusion equation from final data

“…However, under a further assumption ∂ t ρ > 0, [11] employed the strong maximum principles for the elliptic and parabolic equations to derive the uniqueness. There are some other tricks to overcome the ill-posedness of this inverse problem, we refer to [6] and [7] where the diffusion equation was parametrized by a diffusion coefficient and a stability inequality was established except for a countable set of parameters provided that ρ(•, T ) > 0 in Ω, and refer to [3] in which the Carleman estimate was used to recover the source from the measurement of the temperature at fixed time provided that the source is known in an arbitrary subdomain.…”

Inverse problems of determining sources of the fractional partial differential equations

“…As monograph, we should refer to Prilepko, Orlovsky, and Vasin [32]. Moreover see for example Choulli and Yamamoto [6][7][8], Isakov [13], Tikhonov and Eidelman [40] and the references therein for related inverse problems for usual partial differential equations.…”

An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation

“…Uniqueness results for multidimensional inverse problems from a single observation of the solution were first derived by Bukhgeim and Klibanov [14] or Yamamoto [50] when Γ ♯ = Γ, by means of suitable Carleman estimates. For the analysis of inverse coefficients problems with a finite number of observations, based on Carleman estimates, we also refer to, Bellassoued [5,6], Bellassoued, Imanuvilov and Yamamoto [11], Bellassoued and Yamamoto [9,10], Bukhgeim [12], Bukhgeim, Cheng, Isakov and Yamamoto [13], Choulli and Yamamoto [16,17], Imanuvilov and Yamamoto [22,23,24], Isakov [25], Isakov and Yamamoto [26], Kha ȋdarov [31], Klibanov [32], Klibanov and Yamamoto [35], Puel and Yamamoto [43], and Yamamoto [50].…”