An inverse problem for a quasilinear convection–diffusion equation. (September 2022)

• Record Type: Journal Article
• Authors: Feizmohammadi, Ali; Kian, Yavar; Uhlmann, Gunther
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: primary 35R30; Quasilinear parabolic equations -- Nonlinear Fokker–Planck equations -- Inverse problem -- Determination of nonlinear terms; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112921
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the inverse problem of recovering a semilinear diffusion term a ( t, λ ) as well as a quasilinear convection term B ( t, x, λ, ξ ) in a nonlinear parabolic equation ∂ t u − div ( a ( t, u ) ∇ u ) + B ( t, x, u, ∇ u ) ⋅ ∇ u = 0, in ( 0, T ) × Ω, given the knowledge of the flux of the moving quantity associated with different sources applied at the boundary of the domain. This inverse problem that is modeled by the solution dependent parameters a and B has many physical applications related to various classes of cooperative interactions or complex mixing in diffusion processes. Our main result states that, under suitable assumptions, it is possible to fully recover the nonlinear diffusion term a as well as the nonlinear convection term B . The recovery of the diffusion term is based on the idea of solutions to the linearized equation with singularities near the boundary ∂ Ω . Our proof of the recovery of the convection term is based on the idea of higher order linearization to reduce the inverse problem to a density property for certain anisotropic products of solutions to the linearized equation. We show this density property by constructing sufficiently smooth geometric optic solutions concentrating on rays in Ω .