Polynomial stability of operator semigroups. (English) Zbl 1118.47034
The authors study the polynomial decay of classical solutions of linear evolution equations. For bounded strongly continuous semigroup on a Banach space, this property is closely related to polynomial growth estimates of the resolvent of the generator. For systems of commuting normal operators, polynomial decay is characterized in terms of the location of the generator spectrum. The results are applied to systems of coupled wave-type equations.
Reviewer: Gen Qi Xu (Tianjin)
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 34G10 | Linear differential equations in abstract spaces |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 47A10 | Spectrum, resolvent |
Keywords:
asymptotic behaviour; polynomial stability; \(C_{0}\)-semigroups; spectrum; Laplace transform; normal operators; coupled wave equationReferences:
| [1] | Alabau–Boussouira, SIAM J. Control Optim. 41 pp 511– (2002) |
| [2] | Alabau, J. Evol. Equ. 2 pp 127– (2002) |
| [3] | , , and , Vector Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics Vol. 96 (Birkhäuser, Basel, 2001). · Zbl 0978.34001 |
| [4] | Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications Vol. 15 (Birkhäuser, Basel, 1985). |
| [5] | and , Linear Operators. Part II: Spectral Theory (Interscience, New York, 1963). |
| [6] | Operator Matrices and Systems of Evolution Equations, Book manuscript, available at: http://www.fa.unituebingen.de/people/klen/omsee.ps |
| [7] | and , One-parameter Semigroups for Linear Evolution, Graduate Texts in Mathematics Vol. 194 (Springer-Verlag, Berlin, 1999). |
| [8] | Kreiss, Math. Scand. 7 pp 71– (1959) · Zbl 0090.09801 · doi:10.7146/math.scand.a-10563 |
| [9] | and , Hyperbolicity of semigroups and Fourier multipliers, in: Systems, Approximation, Singular Integral Operators, and Related Topics, Proceedings of the International Workshop on Operator Theory and Applications, Bordeaux, 2000, edited by A. A. Boricheva and N. K. Nikolski, Operator Theory: Advance and Applications Vol. 129 (Birkhäuser, Basel, 2001), pp. 341–364. |
| [10] | Lebeau, Arch. Ration. Mech. Anal. 148 pp 179– (1999) |
| [11] | The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory: Advance and Applications Vol. 88 (Birkhäuser, Basel, 1996). |
| [12] | Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, Berlin, 1983). |
| [13] | Functional Analysis (Tata McGraw–Hill, New Delhi, 1974). |
| [14] | Spektraldarstellung linearer Transformationen des Hilbertschen Räumes, Corrected Reprint (Springer-Verlag, Berlin, 1967). |
| [15] | Tcheugoué Tébou, Comm. Partial Differential Equations 23 pp 1839– (1998) |
| [16] | Weis, Proc. Amer. Math. Soc. 124 pp 3663– (1996) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
