Access the full text.
Sign up today, get DeepDyve free for 14 days.
K. Ramdani, Takéo Takahashi, G. Tenenbaum, M. Tucsnak (2005)
A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator.Journal of Functional Analysis, 226
N. Burq (1998)
Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réelActa Mathematica, 180
A. Pazy (1992)
Semigroups of Linear Operators and Applications to Partial Differential Equations, 44
N. Burq, M. Zworski (2004)
Geometric control in the presence of a black boxJournal of the American Mathematical Society, 17
Luc Miller (2004)
Controllability cost of conservative systems: resolvent condition and transmutationJournal of Functional Analysis, 218
W. Arendt, C. Batty (1988)
Tauberian theorems and stability of one-parameter semigroupsTransactions of the American Mathematical Society, 306
Xu Zhang, E. Zuazua (2007)
Long-Time Behavior of a Coupled Heat-Wave System Arising in Fluid-Structure InteractionArchive for Rational Mechanics and Analysis, 184
J. Neerven (1996)
The Asymptotic Behaviour of Semigroups of Linear Operators
Zhuangyi Liu, B. Rao (2005)
Characterization of polynomial decay rate for the solution of linear evolution equationZeitschrift für angewandte Mathematik und Physik ZAMP, 56
T. Eisner, B. Farkas, R. Nagel, András Serény (2007)
Weakly and almost weakly stable C0-semigroupsInternational Journal of Dynamical Systems and Differential Equations, 1
J. Rauch, Xu Zhang, E. Zuazua (2005)
Polynomial decay for a hyperbolic–parabolic coupled systemJournal de Mathématiques Pures et Appliquées, 84
C. Batty (1994)
Asymptotic behaviour of semigroups of operatorsBanach Center Publications, 30
K. Engel, R. Nagel (1999)
One-parameter semigroups for linear evolution equationsSemigroup Forum, 63
J. Korevaar (1982)
On newman’s quick way to the prime number theoremThe Mathematical Intelligencer, 4
A. Bátkai, K. Engel, J. Prüss, R. Schnaubelt (2006)
Polynomial stability of operator semigroupsMathematische Nachrichten, 279
G. Lebeau, L. Robbiano (1997)
Stabilisation de l’équation des ondes par le bordDuke Mathematical Journal, 86
W. Arendt (2002)
Vector-valued laplace transforms and cauchy problemsIsrael Journal of Mathematics, 59
Yu. Lyubich, P. Vu (1988)
Asymptotic stability of linear differential equations in Banach spacesStudia Mathematica, 88
D. Newman (1980)
Simple Analytic Proof of the Prime Number TheoremAmerican Mathematical Monthly, 87
D. Russell, G. Weiss (1994)
A General Necessary Condition for Exact ObservabilitySiam Journal on Control and Optimization, 32
G. Lebeau, E. Zuazua (1999)
Decay Rates for the Three‐Dimensional Linear System of ThermoelasticityArchive for Rational Mechanics and Analysis, 148
G. Vodev (2004)
Local energy decay of solutions to the wave equation for nontrapping metricsArkiv för Matematik, 42
Xiaoyu Fu (2008)
Logarithmic decay of hyperbolic equations with arbitrary boundary dampingarXiv: Analysis of PDEs
K. Phung (2003)
Polynomial decay rate for the dissipative wave equationJournal of Differential Equations, 240
Zhuangyi Liu, B. Rao (2007)
Frequency domain approach for the polynomial stability of a system of partially damped wave equationsJournal of Mathematical Analysis and Applications, 335
Thomas Duyckaerts (2007)
Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interfaceAsymptot. Anal., 51
G. Lebeau (1996)
Equation des Ondes Amorties
R. Chill, Y. Tomilov (2007)
Stability of operator semigroups: ideas and resultsBanach Center Publications, 75
A. Ingham (1935)
On Wiener's Method in Tauberian TheoremsProceedings of The London Mathematical Society
C. Morawetz (1975)
Decay for solutions of the exterior problem for the wave equationCommunications on Pure and Applied Mathematics, 28
N. Burq, M. Hitrik (2006)
Energy decay for damped wave equations on partially rectangular domainsMathematical Research Letters, 14
C. Morawetz, J. Ralston, W. Strauss (1977)
Decay of solutions of the wave equation outside nontrapping obstaclesCommunications on Pure and Applied Mathematics, 30
Let S ( t ) be a bounded strongly continuous semi-group on a Banach space B and – A be its generator. We say that S ( t ) is semi-uniformly stable when S ( t )( A + 1) −1 tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities. In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S ( t )( A + 1) −1 , linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability).
Journal of Evolution Equations – Springer Journals
Published: Nov 1, 2008
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.