The paper deals with the explicit construction and some properties of the fundamental solution \(<E,\cdot>\) of the strictly hyperbolic operator \(L=\epsilon \partial_ t(\partial^ 2_ t-c^ 2_ 1\Delta)+\partial^ 2_ t-c^ 2_ 0\Delta (x\in {\mathbb{R}}^ 3)\), related to the threedimensional wave propagation in dissipative models of thermochemistry, linear viscoelasticity, hereditary electromagnetism. It is proved that \(<E,\cdot>\) is a tempered positive Radon measure induced by a \(C^{\infty}\) rapidly decreasing and positive-value function E. Furthermore, the distributional and the classical initial value problem for L with quite arbitrary data is explicitly solved. The behavior of E when \(t/\epsilon \mapsto \infty\) is also examined and appropriate estimates are achieved.
The author has continued successively the study of the ”wave hierarchy” characterized by L [Ann. Mat. Pura Appl., IV. Ser. 136, 355-389 (1984)]. The explicit fundamental solutions are also constructed when the number n of space dimensions is \(n=1\) or \(n=2\) and the corresponding Cauchy problems \(P_ n\) are solved. As another example, also the solution of the signaling problem is established. Further, various basic aspects of the wave behavior as diffusion, asymptotic properties, maximum principles and the generalized Huygens principle are discussed. Moreover, singular perturbation problems as \(\epsilon \mapsto 0\), with rigorous estimates of the remainder terms uniformly valid for all \(t\geq 0\) are examined.