Well-posedness and stability of the Lagrange representation of the n-D wave equation via boundary triples

TL;DR

This paper establishes well-posedness and semi-uniform stability for the n-D wave equation with complex boundary conditions and variable coefficients using boundary triple theory, extending classical models to more general settings.

Contribution

It introduces a novel analysis of the wave equation with nonlocal and variable coefficients, employing boundary triples to prove well-posedness and stability.

Findings

Proved well-posedness of the wave equation with generalized Laplacian.

Established semi-uniform stability under certain conditions.

Extended classical wave models to include nonlocal operators and complex boundary conditions.

Abstract

We study the Lagrange representation of the wave equation with generalized Laplacian $\operatorname{div} T \nabla$. We allow the coefficients -- the Young modulus $T$ and the density $\rho$ -- to be $\mathrm{L}^{\infty}$ or even nonlocal operators. Moreover, the Lipschitz boundary of the domain $\Omega$ can be split into several parts admitting Dirichlet, Neumann and/or Robin-boundary conditions of displacement, velocity and stress. We show well-posedness of this classical model of the wave equation utilizing boundary triple theory for skew-adjoint operators. In addition we show semi-uniform stability of solutions under slightly stronger assumptions by means of a spectral result.