Conditional stability in determining source terms of semilinear parabolic partial differential equations

TL;DR

This paper investigates the inverse problem of determining an unknown source term in a semilinear parabolic PDE using partial boundary and interior observations, employing advanced analytical techniques to establish conditional stability.

Contribution

It extends Carleman estimate methods to a nonlinear setting for inverse source problems, providing a new approach for stability analysis.

Findings

Achieved conditional Holder stability for the inverse problem.

Extended linear Carleman methods to nonlinear PDEs.

Developed a paradifferential paralinearization technique.

Abstract

We study an inverse source problem for a semilinear parabolic equation in a bounded domain, where the nonlinearity depends on the unknown function and its gradient through a quadratic reaction term and a Burgers-type convection term. From partial boundary observation of the time derivative and its spatial gradient on an open portion of the boundary, together with an interior snapshot of the solution at a fixed time, we aim to recover an unknown spatial source factor. The analysis combines (i) a paradifferential paralinearization in Besov spaces on a short time window, which converts the nonlinear model into a linear parabolic equation with small bounded coefficients, and (ii) a Carleman estimate for the time-differentiated equation, yielding conditional Holder stability. The approach extends the cut-off free Carleman method for linear inverse source problems to a nonlinear setting while keeping the observation geometry unchanged.