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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

Yuri Latushkin 1, and  Valerian Yurov 1,

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.
Keywords: spectral bound, Fourier multipliers., growth bound, evolution semigroups, Gearhart-Prüss Theorem, Convolution.
Mathematics Subject Classification: Primary: 47D06; Secondary: 34D2.
Citation: Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & 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.  Google Scholar

[2]

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

[3]

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

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 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), 2434-2472. 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), 443-471. 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), 981-1030. 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, New York, 1985.  Google Scholar

[11]

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

[12]

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

[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).  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[4]

K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 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), 2434-2472. 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), 443-471. 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), 981-1030. 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, New York, 1985.  Google Scholar

[11]

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

[12]

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

[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).  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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