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Abstrakty
A smooth variation of constants formula for semilinear hyperbolic systems is established using a suitable Banach space X of continuous functions together with its sun dual space [wzór]. It is shown that mild solutions of this variation of constants formula generate a smooth semiflow in X. This proves that the stability of stationary states for the nonlinear flow is determined by the stability of the linearized semigroup.
Wydawca
Czasopismo
Rocznik
Tom
Strony
79--100
Opis fizyczny
Bibliogr. 14 poz.
Twórcy
autor
- Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr, lichtner@wias-berlin.de
Bibliografia
- [1] Cazenave, T., Haraux, A., An Introduction to Semilinear Evolution Equations. Oxford Lecture Ser. Math. Appl. 13, Clarendon Press, Oxford, 1998.
- [2] Clement, Ph., Diekmann, O., Gyllenberg, M., Heijmans, H. J. A. M., Thieme, H. R.. Perturbation theory for dual semigroups, i: The sun-reflexive case, Math. Ann. 277 (1987) 709-725.
- [3] Clement, Ph., Diekmann, O., Gyllenberg, M., Heijmans, H. J. A. M.. Thieme. H. R., Perturbation theory for dual semigroups. iii: Nonlinear Lipschitz continuous perturbations in the sun-reflexive case, in "Volterra Integro-Differential Equations in Banach Spaces and Applications", Pitman Res. Notes Math. Ser. 190. Longman Sci. Tech., Harlow, 1989, 67-89.
- [4] Desch, W., Schappacher, W., Linearized stability for nonlinear semigroups. Differential equations in Banach spaces (Bologna, 1985), Lecture Notes in Math. 1223. Springer, Berlin, 1986, 61-73.
- [5] Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M., Walther, H.-P., Delay Equations, Springer, New York, 1995.
- [6] Gröger, K., Recke, L., Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data. Nonlinear Differential Equations Appl. (NoDEA) 13(3) (2006), 263-285.
- [7] Hille, E., Phillips, R., Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, RI, 1957.
- [8] Krasnoselskij, M. A., Zabreiko, P. P., Pustyl'nik, E. I., Sobolevskii, P. E., Integral Operators in Spaces of Summable Functions, Noordhoff Internat. Publ., Leiden, 1976.
- [9] Lichtner, M., Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems, PhD thesis (2006), pp. 195, http://dochost.rz.huberlin.de/browsing/dissertationen/.
- [10] Lichtner, M., Principle of linearized stability and center manifold theorem for semilinear hyperbolic systems, WIAS preprint 1155 (2006), pp. 30.
- [11] Lichtner, M., Spectral mapping theorem for linear hyperbolic systems, Proc. Amer. Math. Soc. 136 (2008), 2091-2101.
- [12] Lichtner, M., Radziunas, M., Recke, L., Well-posedness, smooth dependence, and centre manifold reduction for a semilinear hyperbolic system from laser dynamics. Math. Methods Appl. Sci. 30 (2006), 931-960.
- [13] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1994.
- [14] van Neerven, J., The Anoint of a. Semigroup of Linear Operators, Lecture Notes in Math. 1529, Springer, Berlin-Heidelberg, 1992.
Typ dokumentu
Bibliografia
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bwmeta1.element.baztech-article-LOD6-0006-0035