Positive solutions of viscoelastic problems

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.