On a partial data inverse problem for the semi-linear wave equation
Boya Liu, Weinan Wang
TL;DR
We address recovering time-dependent nonlinearities of order at least three in a semi-linear wave equation from partial boundary data on a globally hyperbolic Lorentzian manifold with boundary. The approach combines higher-order linearization with Gaussian beam quasimodes that reflect off the boundary, enabling control of inaccessible boundary terms under arbitrarily small measurement sets. The main contributions are a partial data uniqueness result for the cubic nonlinearity $V_3$ and an inductive procedure to recover all higher coefficients $V_m$ ($m\ge4$) on the reachable set ${\mathbb U}$, assuming no conjugate points along null geodesics in ${\mathbb U}$. This demonstrates that nonlinear interactions can compensate for limited data and provides a framework for partial data inverse problems in nonlinear hyperbolic PDEs with minimal geometric restrictions.
Abstract
We show that a partial Dirichlet-to-Neumann map, where the measurement set is arbitrarily small, uniquely determines the time-dependent nonlinearity of order three or higher in a semi-linear wave equation up to natural obstructions on a Lorentzian manifold with boundary. In particular, we do not impose any geometric or size restrictions on the measurement set. The proof relies on the technique of higher order linearization combined with the construction of Gaussian beams with reflections on the boundary.