An inverse problem for semilinear equations involving the fractional Laplacian

TL;DR

This paper investigates inverse problems for heat and wave equations with fractional Laplacians, focusing on recovering nonlinear terms using boundary data, Runge approximation, and unique continuation properties.

Contribution

It introduces a method to recover nonlinear terms in semilinear fractional Laplacian equations from boundary measurements, combining advanced analytical techniques.

Findings

Successfully recovers nonlinear terms from Dirichlet-to-Neumann data

Utilizes Runge approximation for fractional operators

Establishes unique continuation for fractional Laplacian

Abstract

Our work concerns the study of inverse problems of heat and wave equations involving the fractional Laplacian operator with zeroth order nonlinear perturbations. We recover nonlinear terms in the semilinear equations from the knowledge of the fractional Dirichlet-to-Neumann type map combined with the Runge approximation and the unique continuation property of the fractional Laplacian.