An inverse problem for a quasilinear convection--diffusion equation

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 equationgiven 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.

An inverse boundary value problem for isotropic nonautonomous heat flows

“…[12,29]. Indeed, through a first order linearization procedure introduced by Isakov in [15], it is possible to reduce the determination of time-independent coefficients appearing in non-linear equations to the recovery of spacetime dependent coefficients appearing in a linear equation, see also the recent result [11] for further examples of linearizations of inverse problems for nonlinear parabolic equations. Physical motivation aside, (IP) may be viewed as an evolutionary analogue of the well known isotropic Calderón problem, also known as electrical impedance tomography, that is concerned with recovery of a static isotropic electrical conductivity through making voltage and current measurements on the surface of a medium [3].…”