An inverse coefficient problem for a semilinear wave equation by the first order linearization

This paper investigates recovery of an unknown coefficient in a semilinear wave equation defined on a bounded, open, and strictly convex domain in \(\mathbb{R}^{1+n}\) with \(n \ge 2\).

We demonstrate that the unknown coefficient \(q\) appearing in the semilinear wave equation \(\square u + q u^m = 0\) with Neumann boundary conditions can be reconstructed with Hölder stability from the linearized Neumann-to-Dirichlet (NtD) map.

Our approach combines first-order linearization with the Principle of Inclusion-Exclusion (PIE) identity, and employs geometric optics solutions for wave equations in two distinct regimes: the case \(m=2\) with \(q = q(x)\), and the case \(m \ge 3\) with \(q = q(t,x)\).

Furthermore, numerical examples illustrate that the unknown coefficient can also be effectively reconstructed using a neural network-based inversion algorithm within the framework of the least squares method.