Simultaneous recoveries for semilinear parabolic systems

TL;DR

This paper investigates inverse boundary problems for semilinear parabolic systems, demonstrating that both nonlinearities and initial data can be uniquely and stably recovered using boundary measurements, with methods depending on the nonlinearity class.

Contribution

It introduces new techniques for simultaneous recovery of unknown nonlinearities and initial data in semilinear parabolic systems, utilizing control methods and linearization strategies.

Findings

Unique recovery of nonlinearities and initial data from boundary data.

Stable recovery results under certain growth conditions.

Effectiveness of Carleman estimates and CGO solutions depending on nonlinearity class.

Abstract

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general $C^1$ class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special CGO (Complex Geometrical Optics) solutions for the parabolic system.