Inverse problems for a quasilinear strongly damped wave equation arising in nonlinear acoustics

TL;DR

This paper addresses inverse problems for a damped nonlinear wave equation in acoustics, demonstrating unique determination of unknown coefficients from boundary measurements using advanced mathematical techniques.

Contribution

It introduces new uniqueness results for inverse problems involving a Westervelt equation with strong damping and a time-dependent potential, employing complex geometric optics and H"ormander's fundamental solutions.

Findings

Unique determination of potential q and nonlinear coefficient β from boundary data.

Boundary measurements including initial, final, and lateral data suffice for recovery.

Dirichlet-to-Neumann map determines q and β with vanishing initial conditions.

Abstract

We consider inverse problems for a Westervelt equation with a strong damping and a time-dependent potential $q$. We first prove that all boundary measurements, including the initial data, final data, and the lateral boundary measurements, uniquely determine $q$ and the nonlinear coefficient $\beta$. The proof is based on complex geometric optics construction and the approach proposed by Isakov. Further, by considering fundamental solutions supported in a half-space constructed by H\"ormander, we prove that with vanishing initial conditions the Dirichlet-to-Neumann map determines $q$ and $\beta$.