Exponential decay of 2 × 2 operator matrix semigroups

Journal of Computational Analysis and Applications

Volumn 6, Issue 2, 2004, Pages 153-163

  Bátkai, András ^a,b   Engel, Klaus Jochen ^c  

^a EÖTVÖS LORÁND UNIVERSITY   (Hungary)

^b UNIVERSITY OF ROME TOR VERGATA   (Italy)

^c UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

Asymptotic behavior of Co semigroups; Damped wave equations; Operator matrices

Indexed keywords

EID: 33748639075     PISSN: 15211398     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

1. [Ad] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
2. [AR] W. Arendt, A. Rhandi, Perturbation of positive semigroups, Arch. Math. 56 (1991), 107-119.
3. [CR] G. Chen, D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1982), 433-454.
4. [CT] G. Chen, R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. of Math. 136 (1989), 15-55.
5. [CZ1] S. Cox, E. Zuaza, The rate at which energy decays in a damped string, Comm. Partial Differential Equations 19 (1994), 213-243.
6. [CZ2] _, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J. 44 (1995), 545-573.
7. [Da] E. B. Davies, Spectral Theory and Differential Operators, CUP, 1995.
8. [DD] R. Dal Passo, P. De Mottoni, On a paper by A. I. Lee and J. M. Hill, J. Math. Anal. Appl. 106 (1985), 184-187.
9. [EN] K.-J. Engel, R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer-Verlag, 1999.
10. [Fa] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North Holland Mathematical Studies Vol. 108, North Holland, 1985.
11. [Go] G. C. Gorain, Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain, J. Math. Anal. Appl. 216 (1997), 510-520.
12. [Hu] S. -Z. Huang, On energy decay rate of linear damped elastic systems, Tübinger Berichte zur Funktionalanalysis 6 (1996/97), 65-91.
13. [Ho1] A. Holderrieth, Matrix multiplication operators generating one parameter semigroups, Semigroup Forum 42 (1991), 155-166.
14. [Ho2] A. Holderrieth, Multiplicative Perturbations, PhD Dissertation, Tübingen, 1992.
15. [Ka] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980.
16. [Ko] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Wiley, 1994.
17. [Le] G. Lebeau, Equation des ondes amorties, Collection: Algebraic and geometric methods in mathematical physics (A. Boutet de Monvel, V. Marchenko, ed.), Kluwer, 1996, pp. 73-109.
18. [Lo] J. López-Gómez, On the linear damped wave equation, J. Differential Equations 134 (1997), 26-45.
19. [Na] R. Nagel, Operator matrices and reaction-diffusion systems, Rend. Sem. Mat. Fis. Milano 59 (1989), 185-196.
20. [dO] L. A. F. de Oliveira, On reaction-diffusion systems, Electron. J. Differential Equations 1998 (1998), 1-10.
21. [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983.
22. [Re] M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows, Z. Angew. Math. Phys. 45 (1994), 854-865.
23. [Sh] A. Ya. Shklyar, Complete Second Order Linear Differential Equations in Hilbert Spaces, Birkhäuser, 1997.
24. r localizing coefficient, Comm. Partial Differential Equations 23 (1998), 1839-1855.
25. [Tr] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl. 137 (1989), 438-461.
26. [We] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, 1980.
27. [XL] T. J. Xiao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Lect. Notes in Math., vol. 1701, Springer-Verlag, 1998.