Simultaneous stable determination of quasilinear terms for parabolic equations

TL;DR

This paper addresses the inverse problem of simultaneously recovering two classes of quasilinear terms in parabolic equations from boundary data, providing stability estimates that could improve numerical reconstruction methods.

Contribution

It introduces new stability estimates for the simultaneous recovery of quasilinear terms in parabolic equations, combining linearization and singular solutions techniques.

Findings

Derived Lipschitz and Hölder stability estimates.

Applicable to heat conduction and population dynamics models.

Enhances understanding of inverse problems for parabolic PDEs.

Abstract

In this work, we consider the inverse problem of simultaneously recovering two classes of quasilinear terms appearing in a parabolic equation from boundary measurements. It is motivated by several industrial and scientific applications, including the problems of heat conduction and population dynamics, and we study the issue of stability. More precisely, we derive simultaneous Lipschitz and H\"older stability estimates for two separate classes of quasilinear terms. The analysis combines different arguments including the linearization technique with a novel construction of singular solutions and properties of solutions of parabolic equations with nonsmooth boundary conditions. These stability results may be useful for deriving the convergence rate of numerical reconstruction schemes.