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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. [2] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace transforms and Cauchy Problems,", Springer-Verlag, (2011). [3] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Math. Surv. Monogr., 70 (1999). [4] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (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. 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. 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. 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, (1985). [11] B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds,, preprint (2010) , (2010). [12] M. Hieber, Operator valued Fourier multipliers,, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 363. [13] M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers,, in, 215 (2001), 121. [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. [15] Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers,", Systems, 129 (2000), 341. [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. [17] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.

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##### References:
 [1] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications,, Math. Nachr., 186 (1997), 5. 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, (2011). [3] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Math. Surv. Monogr., 70 (1999). [4] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (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. 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. 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. 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, (1985). [11] B. Helffer and J. Sjöstrand, From resolvent bounds to semigroup bounds,, preprint (2010) , (2010). [12] M. Hieber, Operator valued Fourier multipliers,, Progress in Nonlinear Differential Equations and Their Applications, 35 (1999), 363. [13] M. Hieber, A characterization of the growth bound of a semigroup via Fourier multipliers,, in, 215 (2001), 121. [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. [15] Y. Latushkin and R. Shvydkoy, "Hyperbolicity of Semigroups and Fourier Multipliers,", Systems, 129 (2000), 341. [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. [17] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1.
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