• Previous Article
    Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
  • DCDS Home
  • This Issue
  • Next Article
    Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace
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 & Continuous Dynamical Systems - A, 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. doi: 10.1002/mana.3211860102. Google Scholar

[2]

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

[3]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Math. Surv. Monogr., 70 (1999). Google Scholar

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar

[5]

A. Ghazaryan, A. Hoffman, K. Promislov and S. Schecter, Private, Communications., (). Google Scholar

[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. doi: 10.1137/100786204. Google Scholar

[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. doi: 10.1512/iumj.2011.60.4069. Google Scholar

[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. doi: 10.1007/s00205-010-0358-y. Google Scholar

[9]

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

[10]

J. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Univ. Press, (1985). Google Scholar

[11]

B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds,, preprint (2010) , (2010). Google Scholar

[12]

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

[13]

M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers,, in, 215 (2001), 121. Google Scholar

[14]

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

[15]

Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers,", Systems, 129 (2000), 341. Google Scholar

[16]

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

[17]

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

show all references

References:
[1]

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

[2]

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

[3]

C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Math. Surv. Monogr., 70 (1999). Google Scholar

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar

[5]

A. Ghazaryan, A. Hoffman, K. Promislov and S. Schecter, Private, Communications., (). Google Scholar

[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. doi: 10.1137/100786204. Google Scholar

[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. doi: 10.1512/iumj.2011.60.4069. Google Scholar

[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. doi: 10.1007/s00205-010-0358-y. Google Scholar

[9]

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

[10]

J. Goldstein, "Semigroups of Linear Operators and Applications,", Oxford Univ. Press, (1985). Google Scholar

[11]

B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds,, preprint (2010) , (2010). Google Scholar

[12]

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

[13]

M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers,, in, 215 (2001), 121. Google Scholar

[14]

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

[15]

Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers,", Systems, 129 (2000), 341. Google Scholar

[16]

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

[17]

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

[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]

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

[3]

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

[4]

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

[5]

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.

[6]

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.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Lassi Roininen, Markku S. Lehtinen, Petteri Piiroinen, Ilkka I. Virtanen. Perfect radar pulse compression via unimodular fourier multipliers. Inverse Problems & Imaging, 2014, 8 (3) : 831-844. doi: 10.3934/ipi.2014.8.831

[16]

Qing Hong, Guorong Hu. Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3103-3120. doi: 10.3934/cpaa.2019139

[17]

Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037

[18]

Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial & Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647

[19]

J. De Beule, K. Metsch, L. Storme. Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound. Advances in Mathematics of Communications, 2008, 2 (3) : 261-272. doi: 10.3934/amc.2008.2.261

[20]

Florent Foucaud, Tero Laihonen, Aline Parreau. An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid. Advances in Mathematics of Communications, 2014, 8 (1) : 35-52. doi: 10.3934/amc.2014.8.35

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]