On the determination of nonlinear terms appearing in semilinear hyperbolic equations

TL;DR

This paper establishes the unique determination of nonlinear terms in semilinear hyperbolic equations on Riemannian manifolds using boundary measurements, advancing inverse problem solutions in geometric PDEs.

Contribution

It proves the first results on uniquely recovering general nonlinear terms in semilinear hyperbolic equations on Riemannian manifolds from boundary data.

Findings

Unique recovery of nonlinear term $F(t,x,u)$ on boundary and inside manifold.

Applicable to 2D and 3D Riemannian manifolds.

Boundary measurements suffice for nonlinear term determination.

Abstract

We consider the inverse problem of determining a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary $(M,g)$ of dimension $n=2,3$. We prove results of unique recovery of the nonlinear term $F(t,x,u)$, appearing in the equation $\partial_t^2u-\Delta_gu+F(t,x,u)=0$ on $(0,T)\times M$ with $T>0$, from some partial knowledge of the solutions $u$ on the boundary of the time-space cylindrical manifold $(0,T)\times M$ or on the lateral boundary $(0,T)\times\partial M$. We determine the expression $F(t,x,u)$ both on the boundary $x\in\partial M$ and inside the manifold $x\in M$.