Plane-wave analysis of a hyperbolic system of equations with relaxation in $\mathbb{R}^{d}$

TL;DR

This paper analyzes a multi-dimensional wave equation with memory effects in viscoelastic materials, demonstrating well-posedness and long-term stability through plane-wave analysis and hyperbolic system theory.

Contribution

It shows that the relaxation system fits a non-strictly hyperbolic framework satisfying Majda's block structure, with new stability results for systems with dissipative memory.

Findings

The system is well-posed with real, uniformly diagonalizable eigenvalues.

Long-time stability is achieved when the memory term dissipates energy.

The model satisfies Majda's block structure condition.

Abstract

We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.