Positivity of Green's functions for a class of partial integro-differential equations including viscoelasticity

Abstract

Positivity of Green's functions is established for a class of linear partial integro-differential equations in d ≤ 3 space dimensions with Volterra kernels satisfying appropriate conditions. In particular it is assumed that the Volterra kernel is completely monotone. This class includes the equations of motion of scalar hereditary viscoelasticity along with the extreme cases of elasticity and Newtonian viscosity. The positivity of Green's functions is also established under a less restrictive assumption. Positivity results are extended to a class of systems of partial integro-differential equations with completely monotone matrix-valued Volterra kernels. Positivity for longitudinal and transverse modes of 3D isotropic viscoelasticity is also established. Weaker results are obtained for higher space dimensions.