Summary
The paper deals with a rigorous analysis of the «wave hierarchie» related to the operator Ln quoted in the title. Whatever the number n=1, 2, 3 of space dimensions may be, the fundamental solutions En are constructed. These distributions are tempered positive Radon measures associated with positive value functions which have numerous basic properties. So the Cauchy problemP n(n=1,2,3) with quite arbitrary data is explicitly solved. As another example, also the solution of the signaling problem ℋ is established. Then, various basic aspects of the wave behavior such as diffusion, asymptotic properties, maximum principles and the generalized Huyghens principle are evaluated. Moreover, singular perturbation problems as ɛ→ 0, with estimates of the remainder terms uniformly valid for all t⩾0, are discussed too.
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References
G. B. Whitham,Some comments on wave propagation and shock wave structure with applications to magnetohydrodynamics, Comm. on Pure and Appl. Math., Vol. XII (1959), pp. 113–158.
G. B.Whitham,Linear and Nonlinear Waves, J. Wiley & Sons, 1974.
W. Lick,Wave propagation in real gases, Advances in Applied Mechanics, Vol. 10 (1967) pp. 1–72.
J. F.Clarke - M.McChesney,Dynamics of Relaxing Gases, Butterworts, 1976.
H. Kolsky,Stress Waves in Solids, Dover, New-York, 1968.
C.Hunter,Viscoelastie Waves, in Progress in Solid Mechanics, Vol. I, North-Holland, (1964), pp. 1–57.
J. A. Morrison,Waves propagation in rods of Voigt material and visco-elastic material with threeparameter models, Quaterly of Applied Mathematics, Vol. XIV (1956), pp. 153–169.
M. Fabrizio,Sull'equazione di propagazione nei mezzi viscoelastici, Ist. Lombardo Accad. Sci. Lett. Rend. A,102 (1968), pp. 437–441.
D. Graffi,Mathematical models and waves in linear Viscoelasticity, Euromec 127 on «Wave propagation in viscoelastic media», Taormina, 1980. Pitman Adv. Publ. Pr., Research Notes in Math.,52 (1982), pp. 1–27.
G. Crupi,Sulle onde piane magneto-idrodinamiche, Boll. Un. Mat. Ital., (3), Vol. 12 (1957), pp. 439–442.
D. Graffi,Sulla propagazione nei mezzi dispersivi, Ann. Mat. Pura e Appl., (4)60 (1963), pp. 173–196.
A. Donato -D. Fusco,Su alcune proprietà di un modello matematico atto a descrivere fenomeni ereditari in un dielettrico non lineare, Atti Acc. Peloritana dei Pericolanti, Classe I, Sc. Fis. Mat. e Nat.,59 (1981), pp. 149–162.
B. T. Chu,Stress waves in isotropic linear viscoelatic material, Journal de Mécanique, Vol. I, n. 4 (1961), pp. 439–461.
B. J. Matkowsky -E. L. Reiss,On the Asymptotic Theory of Dissipative Wave Motion, Archive for Rational Mechanics and Analysis, Vol. 42 (1971), pp. 194–212.
F. Mainardi -G. Turchetti,Wave front expansions for transient viscoelastic waves, Mech. Res. Comm.,2 (1975), pp. 107–112.
P. Renno,Sulla soluzione fondamentale di un operatore iperbolico della termochimica tridimensionale, Rend. Accademia Nazionale dette Scienze detta dei XL,98, Vol. IV, fasc. 4 (1979–80), pp. 43–62.
C.Cercignani,Wave propagation according to a third hyperbolic equation arising in viscoelasticity, EUROMECH,127, Taormina, April 14–18 (1980).
P. Renno,On the theory of dissipative three-dimensional wave motion, Mechanics Research Communications, Vol. 8 (2) (1981), pp. 83–92.
A.Erdelyi - W.Magnus - F.Oberhettinger - F. G.Tricomi,Tables of Integral Transforms, Vol. I, Mc Graw-Hill Book Co., 1954.
G. N. Watson,Theory of Bessel Functions, Cambridge, Univ. Press., 1958.
J. L.Lions,Perturbations Singulières dans les Problèmes aux Limites et en Contröle Optimal, Lectures Notes in Mathematics, n. 323, Springer-Verlag, 1973.
Z.Szmydt,Fourier Transformation and Linear Differential Equations, D. Reidel Publ. Company, 1977.
M.Schechter,Modern Methods in Partial Differential Equations, Mc Graw-Hill Book Co., 1977.
F.John,Partial Differential Equations, Springer-Verlag, Third Ed., 1978.
W.Eckhaus,Asymptotic Analysis of Singular Perturbations, North-Holland, 1979.
J.Kevorkian - J. D.Cole,Perturbation Methods in Applied Mathematics, Springer-Verlag, 1981.
P.Renno,On an inversion formula of certain Laplace transforms in dissipative wave propagation, Atti Acc. Naz. Lincei Rend., Vol. LXXI, 2° sem., fasc. 6 (1981).
P.Renno,Singular perturbation problems for the operator ɛ∂t(∂t/2-a1/2Δ + ∂t/2 - a0/2Δ, General lecture at the Congress «Onde e Stabilità nei mezzi continui» Catania 1981, Quaderni del C.N.R., 1982.
B.D'Acunto - A.D'Anna - P.Renno,On the motion of a viscoelastic solid in presence of a rigid wall, Part I - Journal of Applied Mathematics and Physics (ZAMP), Vol. 34 (1983).
M. H.Protter - H. F.Weimberger,Maximum principles in differential equations, Prentice-Hall, Inc., 1967.
P.Renno,On some viscoelastic models, Atti Acc. Naz. Lincei Rend.,75, 2° sem., fasc. 6 (1983).
P.Renno,Wave hierarchies in linear dissipative media, General Lecture at Second Congress on «Onde e Stabilità nei mezzi continui», Rende (Cosenza), 6–11 Giugno 1983.
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Some results of this paper have been communicated in a General Lecture at the Congress «Onde e Stabilità nei Mezzi Continui» held at Catania in November 1981.
This research was partially supported by M.P.I. and C.N.R. (Gruppo Fisica Matematica)
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Renno, P. On a wave theory for the operator ɛ∂t(∂ 2 t −c 21 Δn)+∂ 2 t −c 20 Δ n . Annali di Matematica pura ed applicata 136, 355–389 (1984). https://doi.org/10.1007/BF01773390
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DOI: https://doi.org/10.1007/BF01773390