Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

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

Pages: 61 – 79

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n1.a3

Authors

Maarten V. de Hoop (Department of Computational and Applied Mathematics, Rice University, Houston, Texas, U.S.A.)

Jian-Guo Liu (Departments of Mathematics and Physics, Duke University, Durham, North Carolina, U.S.A.)

Peter A. Markowich (Applied Mathematics and Computer Science Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia; and Faculty of Mathematics, University of Vienna, Austria)

Nail S. Ussembayev (Applied Mathematics and Computer Science Program, CEMSE Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

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. Wellposedness 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.

Keywords

characteristic fields of constant multiplicity, eigenvalues, viscoelasticity, memory effect, Zener model, stability, energy methods

2010 Mathematics Subject Classification

35B35, 35L40, 74D05

Full Text (PDF format)

M.V.d.H. gratefully acknowledges support from the Simons Foundation under the MATH+X program, the National Science Foundation under grant DMS-1559587, and the corporate members of the Geo-Mathematical Group at Rice University.

J.-G.L. is supported by the National Science Foundation under grant DMS-1812573 and KI-Net RNMS11-07444.

Received 14 March 2018

Accepted 5 October 2018

Published 30 May 2019