Spectral mapping theorem for linear hyperbolic systems
HTML articles powered by AMS MathViewer
- by Mark Lichtner PDF
- Proc. Amer. Math. Soc. 136 (2008), 2091-2101 Request permission
Abstract:
We show high frequency resolvent and spectral estimates and prove the spectral mapping theorem for linear hyperbolic systems in one space dimension.References
- Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR 1707332, DOI 10.1090/surv/070
- Michael Renardy, Spectrally determined growth is generic, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2451–2453. MR 1328372, DOI 10.1090/S0002-9939-96-03417-X
- Michael Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows, Z. Angew. Math. Phys. 45 (1994), no. 6, 854–865. MR 1306936, DOI 10.1007/BF00952081
- Herbert Koch and Daniel Tataru, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations 20 (1995), no. 5-6, 901–937. MR 1326911, DOI 10.1080/03605309508821119
- A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics reported: expositions in dynamical systems, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 1, Springer, Berlin, 1992, pp. 125–163. MR 1153030
- Peter W. Bates and Christopher K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38. MR 1000974
- Peter W. Bates, Kening Lu, and Chongchun Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. 135 (1998), no. 645, viii+129. MR 1445489, DOI 10.1090/memo/0645
- Peter W. Bates, Kening Lu, and Chongchun Zeng, Persistence of overflowing manifolds for semiflow, Comm. Pure Appl. Math. 52 (1999), no. 8, 983–1046. MR 1686965, DOI 10.1002/(SICI)1097-0312(199908)52:8<983::AID-CPA4>3.3.CO;2-F
- F. Gesztesy, C. K. R. T. Jones, Y. Latushkin, and M. Stanislavova, A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2000), no. 1, 221–243. MR 1777032, DOI 10.1512/iumj.2000.49.1838
- Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385–394. MR 461206, DOI 10.1090/S0002-9947-1978-0461206-1
- Jan Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847–857. MR 743749, DOI 10.1090/S0002-9947-1984-0743749-9
- Aloisio Freiria Neves, Hermano de Souza Ribeiro, and Orlando Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal. 67 (1986), no. 3, 320–344. MR 845461, DOI 10.1016/0022-1236(86)90029-7
- Michael Renardy, On the type of certain $C_0$-semigroups, Comm. Partial Differential Equations 18 (1993), no. 7-8, 1299–1307. MR 1233196, DOI 10.1080/03605309308820975
- M. Lichtner, Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems, PhD thesis, 2006, 167 pp.; http://dochost.rz.hu-berlin.de/browsing/dissertationen/
- —, Principle of linearized stability and center manifold theorem for semilinear hyperbolic systems, WIAS preprint No. 1155, 2006, 28 pp.
- —, Variation of constants formula for hyperbolic systems, WIAS preprint No. 1212, 2007, 19 pp.
- M. Radziunas, Numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers, Phys. D 213 (2006), no. 1, 98–112. MR 2186586, DOI 10.1016/j.physd.2005.11.003
- Aloisio Freiria Neves and Xiao-Biao Lin, A multiplicity theorem for hyperbolic systems, J. Differential Equations 76 (1988), no. 2, 339–352. MR 969429, DOI 10.1016/0022-0396(88)90079-4
- Zheng-Hua Luo, Bao-Zhu Guo, and Omer Morgul, Stability and stabilization of infinite dimensional systems with applications, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1999. MR 1745384, DOI 10.1007/978-1-4471-0419-3
- B. Z. Guo and G. Q. Xu, On Basis property of a hyperbolic system with dynamic boundary condition, Differential Integral Equations, 18 (2005) no. 1, 35–60.
Additional Information
- Mark Lichtner
- Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
- Email: lichtner@wias-berlin.de
- Received by editor(s): March 13, 2007
- Published electronically: February 14, 2008
- Additional Notes: This work has been supported by DFG Research Center Matheon, ‘Mathematics for key technologies’ in Berlin.
- Communicated by: Carmen C. Chicone
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2091-2101
- MSC (2000): Primary 47D03, 47D06, 34D09, 35P20; Secondary 37L10, 37D10
- DOI: https://doi.org/10.1090/S0002-9939-08-09181-8
- MathSciNet review: 2383515