Geodesic
Qualitative properties of structurally damped wave models
Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 84-106 
S. Matthes; M. Reissig. Qualitative properties of structurally damped wave models. Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 84-106. http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a7/
@article{EMJ_2013_4_3_a7,
     author = {S. Matthes and M. Reissig},
     title = {Qualitative properties of structurally damped wave models},
     journal = {Eurasian mathematical journal},
     pages = {84--106},
     year = {2013},
     volume = {4},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a7/}
}
TY  - JOUR
AU  - S. Matthes
AU  - M. Reissig
TI  - Qualitative properties of structurally damped wave models
JO  - Eurasian mathematical journal
PY  - 2013
SP  - 84
EP  - 106
VL  - 4
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a7/
LA  - en
ID  - EMJ_2013_4_3_a7
ER  - 
%0 Journal Article
%A S. Matthes
%A M. Reissig
%T Qualitative properties of structurally damped wave models
%J Eurasian mathematical journal
%D 2013
%P 84-106
%V 4
%N 3
%U http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a7/
%G en
%F EMJ_2013_4_3_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

Qualitative properties such as precise decay estimates for energies of higher order and smoothing effects for a special class of structurally damped wave models are investigated in this paper. The main tools are diagonalization techniques and the theory of zones in the extended phase space. Some special effects for visco-elastic damped wave models are explained. These models are limit cases of structurally damped wave models interpolating between classical damped waves and visco-elastic waves.

[1] G. Avalos, I. Lasiecka, “Optimal blowup rates for the minimal energy null control for the structurally damped abstract wave equation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 601–616 | MR | Zbl

[2] F. Bucci, “A Dirichlet boundary control problem for the strongly damped wave equation”, SIAM J. Control Optim., 30 (1992), 1092–1100 | DOI | MR | Zbl

[3] G. Chen, D. Russel, “A mathematical model for linear elastic systems with structural damping”, Quart. J. of Appl. Math., 39 (1982), 433–454 | MR | Zbl

[4] S. P. Chen, R. Triggiani, “Proof of extensions of two conjectures on structural damping for elastic systems”, Pacific J. Math., 136 (1989), 15–55 | DOI | MR | Zbl

[5] S. P. Chen, R. Triggiani, “Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications”, J. Differential Equations, 88 (1990), 279–293 | DOI | MR | Zbl

[6] I. Lasiecka, L. Pandolfi, R. Triggiani, “A singular control approach to highly damped second-order abstract equations and applications”, Appl. Math. Optim., 36 (1997), 67–107 | MR | Zbl

[7] X. Lu, M. Reissig, “Rates of decay for structural damped models with decreasing in time coefficients”, Int. J. of Dynamical Systems and Differential Equations, 2:1–2 (2009), 21–55 | MR | Zbl

[8] S. Matthes, Qualitative properties of structurally damped wave models, Master's thesis, TU Bergakademie Freiberg, 2011, 81 pp.

[9] M. Reissig, “Rates of decay for structural damped models with strictly increasing in time coefficients”, Contemporary Mathematics, 554 (2011), 187–206 | DOI | MR | Zbl

[10] Y. Shibata, “On the rate of decay of solutions to linear viscoelastic equation”, Math. Meth. Appl. Sci., 23 (2000), 203–226 | 3.0.CO;2-M class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[11] G. F. Webb, “Existence and asymptotic behaviour for a strongly damped non-linear wave equation”, Can. J. Math., 32 (1980), 631–643 | DOI | MR | Zbl

[12] J. Wirth, “Wave equations with time-dependent dissipation. I. Non-effective dissipation”, Journal of Differential Equations, 222:2 (2006), 487–514 | DOI | MR | Zbl

[13] J. Wirth, “Wave equations with time-dependent dissipation. II. Effective dissipation”, Journal of Differential Equations, 232:2 (2007), 74–103 | DOI | MR | Zbl

[14] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple Characteristics. Micro-Local Approach, Math. Topics, 12, Akademie Verlag, Berlin, 1997 | MR | Zbl