Optimal Runge approximation for nonlocal wave equations and unique determination of polyhomogeneous nonlinearities

TL;DR

This paper establishes a Runge approximation property for nonlocal wave equations, enabling unique identification of nonlinearities, and extends inverse problem results in nonlocal PDEs.

Contribution

It introduces a Runge approximation in nonlocal wave equations and applies it to prove unique determination of polyhomogeneous nonlinearities.

Findings

Runge approximation holds in the nonlocal wave setting.

Unique determination of nonlinearities with polyhomogeneous structure.

Extension of Calderón problem results to nonlocal wave equations.

Abstract

The main purpose of this article is to establish the Runge-type approximation in $L^2(0,T;\widetilde{H}^s(\Omega))$ for solutions of linear nonlocal wave equations. To achieve this, we extend the theory of very weak solutions for classical wave equations to our nonlocal framework. This strengthened Runge approximation property allows us to extend the existing uniqueness results for Calder\'on problems of linear and nonlinear nonlocal wave equations in our earlier works. Furthermore, we prove unique determination results for the Calder\'on problem of nonlocal wave equations with polyhomogeneous nonlinearities.