Stability of the determination of a time-dependent coefficient in parabolic equations

TL;DR

This paper proves a Lipschitz stability estimate for identifying a time-dependent coefficient in parabolic equations from boundary data, extending results to semi-linear cases, which enhances the understanding of inverse problems in PDEs.

Contribution

The paper introduces a Lipschitz stability estimate for a time-dependent coefficient in parabolic equations, including semi-linear cases, advancing inverse problem analysis.

Findings

Lipschitz stability estimate established for linear parabolic inverse problem

Extension of stability results to semi-linear parabolic equations

Enhanced understanding of coefficient determination from boundary data

Abstract

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation in changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(t,x,\sigma(t),u(x,t))$.