An inverse problem for a quasilinear convection--diffusion equation

TL;DR

This paper addresses the inverse problem of uniquely recovering nonlinear diffusion and convection terms in a parabolic PDE from boundary flux data, with implications for complex diffusion processes.

Contribution

It introduces a method to fully recover nonlinear diffusion and convection terms using linearization and geometric optics solutions, advancing inverse problem techniques for nonlinear PDEs.

Findings

Full recovery of diffusion term $a(t,)$

Full recovery of convection term $\u211b(t,x,, abla u)$

Use of higher order linearization and geometric optics solutions

Abstract

We study the inverse problem of recovering a semilinear diffusion term $a(t,\lambda)$ as well as a quasilinear convection term $\mathcal B(t,x,\lambda,\xi)$ in a nonlinear parabolic equation $$\partial_tu-\textrm{div}(a(t,u) \nabla u)+\mathcal B(t,x,u,\nabla u)\cdot\nabla u=0, \quad \mbox{in}\ (0,T)\times\Omega,$$ 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 $\mathcal B$ 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 $\mathcal B$. The recovery of the diffusion term is based on the idea of solutions to the linearized equation with singularities near the boundary $\partial \Omega$. 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 $\Omega$.