Stability estimates for  semigroups on Banach spaces

Yuri Latushkin; Valerian Yurov

doi:10.3934/dcds.2013.33.5203

Article Contents

2013, Volume 33, Issue 11&12: 5203-5216. Doi: 10.3934/dcds.2013.33.5203

This issue Previous Article Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system Next Article Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace

Stability estimates for semigroups on Banach spaces

Yuri Latushkin^1, and
Valerian Yurov^1,

1.
Department of Mathematics, University of Missouri, Columbia, MO 65211

Received: July 2011

Revised: July 2011

Published: November 2013

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 47D06; Secondary: 34D20.

Citation:

Related Papers

Cited by

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.