  Using the method of a priori energy estimates, energy dissipation is proved
for the class of hereditary fractional wave equations, obtained through the
system of equations consisting of equation of motion, strain, and fractional
order constitutive models, that include the distributed-order constitutive law
in which the integration is performed from zero to one generalizing all linear
constitutive models of fractional and integer orders, as well as for the
thermodynamically consistent fractional Burgers models, where the orders of
fractional differentiation are up to the second order. In the case of non-local
fractional wave equations, obtained using non-local constitutive models of
Hooke- and Eringen-type in addition to the equation of motion and strain, a
priori energy estimates yield the energy conservation, with the reinterpreted
notion of the potential energy.
