Positive solutions of viscoelastic problems

TL;DR

This paper proves that scalar and vector wave solutions in viscoelastic media with certain properties inherit the sign of their sources, extending to higher dimensions and weaker conditions.

Contribution

It establishes positivity results for scalar and vector viscoelastic wave solutions under broad conditions, including higher dimensions and relaxed assumptions.

Findings

Scalar wave solutions inherit source sign in 1-3D

Positivity extends to vector-valued waves in isotropic media

Results hold under weaker hypotheses

Abstract

In 1,2 or 3 dimensions a scalar wave excited by a non-negative source in a viscoelastic medium with a non-negative relaxation spectrum or a Newtonian response or both combined inherits the sign of the source. The key assumption is a constitutive relation which involves the sum of a Newtonian viscosity term and a memory term with a completely monotone relaxation kernel. In higher-dimensional spaces this result holds for sufficiently regular sources. Two positivity results for vector-valued wave fields including isotropic viscoelasticity are also obtained. Positivity is also shown to hold under weakened hypotheses.