• Previous Article
    Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace
  • DCDS Home
  • This Issue
  • Next Article
    Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
November  2013, 33(11&12): 5203-5216. doi: 10.3934/dcds.2013.33.5203

Stability estimates for semigroups on Banach spaces

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States

Received  July 2011 Revised  July 2011 Published  May 2013

For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.
Citation: Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203
References:
[1]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace transforms and Cauchy Problems," Springer-Verlag, New York, 2011.

[3]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Math. Surv. Monogr., 70, AMS, Providence, 1999.

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.

[5]

A. Ghazaryan, A. Hoffman, K. Promislov and S. Schecter, Private Communications.

[6]

A. Ghazaryan, Y. Latushkin and S. Schecter, Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models, SIAM J. Mathematical Analysis, 42 (2010), 2434-2472. doi: 10.1137/100786204.

[7]

A. Ghazaryan, Y. Latushkin and S. Schecter, Stability of traveling waves for degenerate systems of reaction diffusion equations, Indiana University Math. J., 60 (2011), 443-471. doi: 10.1512/iumj.2011.60.4069.

[8]

A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in one-dimensional solids, Archive Rational Mech. Anal., 198 (2010), 981-1030. doi: 10.1007/s00205-010-0358-y.

[9]

A. Ghazaryan, Y. Latushkin, S. Schecter and V. Yurov, Spectral mapping results for degenerate systems, (In preparation).

[10]

J. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.

[11]

B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds, preprint (2010) arXiv:1001.4171v1.

[12]

M. Hieber, Operator valued Fourier multipliers, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 363-380.

[13]

M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers, in "Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), 121-124, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, (2001).

[14]

Y. Latushkin and F. Räbiger, Operator valued Fourier multipliers and stability of strongly continuous semigroups, Integral Eqns. Oper. Theory, 51 (2005), 375-394. doi: 10.1007/s00020-004-1349-x.

[15]

Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers," Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), 341-363, Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001.

[16]

J. M. A. M. van Neerven, "The Asymptotic Behavior of Semigroups of Linear Operators," Oper. Theory Adv. Appl., 88, Birkhäuser-Verlag, 1996. doi: 10.1007/978-3-0348-9206-3.

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

show all references

References:
[1]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace transforms and Cauchy Problems," Springer-Verlag, New York, 2011.

[3]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Math. Surv. Monogr., 70, AMS, Providence, 1999.

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 2000.

[5]

A. Ghazaryan, A. Hoffman, K. Promislov and S. Schecter, Private Communications.

[6]

A. Ghazaryan, Y. Latushkin and S. Schecter, Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models, SIAM J. Mathematical Analysis, 42 (2010), 2434-2472. doi: 10.1137/100786204.

[7]

A. Ghazaryan, Y. Latushkin and S. Schecter, Stability of traveling waves for degenerate systems of reaction diffusion equations, Indiana University Math. J., 60 (2011), 443-471. doi: 10.1512/iumj.2011.60.4069.

[8]

A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in one-dimensional solids, Archive Rational Mech. Anal., 198 (2010), 981-1030. doi: 10.1007/s00205-010-0358-y.

[9]

A. Ghazaryan, Y. Latushkin, S. Schecter and V. Yurov, Spectral mapping results for degenerate systems, (In preparation).

[10]

J. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Univ. Press, New York, 1985.

[11]

B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds, preprint (2010) arXiv:1001.4171v1.

[12]

M. Hieber, Operator valued Fourier multipliers, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 363-380.

[13]

M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers, in "Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), 121-124, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, (2001).

[14]

Y. Latushkin and F. Räbiger, Operator valued Fourier multipliers and stability of strongly continuous semigroups, Integral Eqns. Oper. Theory, 51 (2005), 375-394. doi: 10.1007/s00020-004-1349-x.

[15]

Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers," Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), 341-363, Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001.

[16]

J. M. A. M. van Neerven, "The Asymptotic Behavior of Semigroups of Linear Operators," Oper. Theory Adv. Appl., 88, Birkhäuser-Verlag, 1996. doi: 10.1007/978-3-0348-9206-3.

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[1]

Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175

[2]

Luís Simão Ferreira. A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. Kinetic and Related Models, 2022, 15 (1) : 91-117. doi: 10.3934/krm.2021045

[3]

Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209

[4]

Mustapha Mokhtar-Kharroubi. On spectral gaps of growth-fragmentation semigroups with mass loss or death. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1293-1327. doi: 10.3934/cpaa.2022019

[5]

Mustapha Mokhtar-Kharroubi, Jacek Banasiak. On spectral gaps of growth-fragmentation semigroups in higher moment spaces. Kinetic and Related Models, 2022, 15 (2) : 147-185. doi: 10.3934/krm.2021050

[6]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97

[7]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems and Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239

[8]

Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3897-3912. doi: 10.3934/dcdsb.2021210

[9]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electronic Research Announcements, 2001, 7: 28-36.

[10]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements, 2001, 7: 17-27.

[11]

Gang Wang, Deng-Ming Xu, Fang-Wei Fu. Constructions of asymptotically optimal codebooks with respect to Welch bound and Levenshtein bound. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021065

[12]

Mikko Kaasalainen. Dynamical tomography of gravitationally bound systems. Inverse Problems and Imaging, 2008, 2 (4) : 527-546. doi: 10.3934/ipi.2008.2.527

[13]

Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467

[14]

Z.G. Feng, K.L. Teo, Y. Zhao. Branch and bound method for sensor scheduling in discrete time. Journal of Industrial and Management Optimization, 2005, 1 (4) : 499-512. doi: 10.3934/jimo.2005.1.499

[15]

Marcin Dumnicki, Łucja Farnik, Halszka Tutaj-Gasińska. Asymptotic Hilbert polynomial and a bound for Waldschmidt constants. Electronic Research Announcements, 2016, 23: 8-18. doi: 10.3934/era.2016.23.002

[16]

Miklós Horváth, Márton Kiss. A bound for ratios of eigenvalues of Schrodinger operators on the real line. Conference Publications, 2005, 2005 (Special) : 403-409. doi: 10.3934/proc.2005.2005.403

[17]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055

[18]

John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

[19]

Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165

[20]

Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]