Abstract
In this paper, the first of a bipartite work, we consider an abstract, nonautonomous system of evolution equations of hyperbolic type, related to semilinear wave equations. Theorem 1 states that under certain assumptions the system admits a global center manifold, or equivalently a global decoupling function which is continuously differentiable with respect to its arguments, among which timet occurs. The difficult proof is presented in part II, i.e. the continuation of the present paper. For purposes of applications a local version of Theorem 1 is proved, i.e. the local center manifold Theorem 2. We obtain a series of applications both to abstract, nonautonomous wave equations and to concrete nonautonomous, semilinear wave equations subject to Neumann and Dirichlet boundary conditions.
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References
A. Adams,Sobolev Spaces. Academic Press, New York 1975.
P. Bates and C. K. Jones,Invariant manifolds for semilinear partial differential equations. Dynamics Reported, Vol. 2, 1–38 (1989).
L. Bers, F. John and M. Schechter,Partial Differential Equations. John Wiley, New York 1964.
J. Carr,Applications of Center Manifold Theory. Appl. Math. Sci. 35, Springer, Berlin 1981.
W. Coppel,Dichotomies in Stability Theory. Lect. Notes in Math. 629, Springer, Berlin 1970.
W. Eleonski, N. Kulagin, N. Novozhilova and V. Silin,Asymptotic expansions and qualitative analysis affinite dimensional models in non-linear field theory. Theoreticheskaya Mathematichskaya Fiska60 No. 2, 137–157 (1984).
G. Fischer,Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen. Math. Nachrichten115, 703–728 (1984).
J. Hadamard,Sur l'itération et les solutions asymptotiques des equations différentielles. Bull. Soc. Math. France29, 224–228 (1901).
D. Henry,Geometrie Theory of Semilinear Parabolic Equations. Lect. Notes in Math. 840, Springer, Berlin 1981.
S. Jawad,Die klassisch reguläre Lösbarkeit des Rand-Anfangswertproblems nichtlinearer Wellengleichungen mit einer Randbedingung vom gemischten Typ. Math. Z.199, 4, 479–490 (1988).
K. Kirchgässner,Wave-solutions of reversible systems and applications. J. Diff. Eq.45, 113–127 (1982).
K. Kirchgässner and J. Scheuerle,On the bounded solutions of a semi-linear equation on a strip. J. Diff. Eq.32, 119–148 (1979).
M. Kruskal and H. Segur,Nonexistence of smallamplitude breather solutions in ϕ 4-theory. Phys. Rev. Lett.58, no. 8, 747–750 (1987).
C. A. Ladyzenskaya,The Boundary Value Problems of Mathematical Physics. Appl. Math. Sci. 49, Springer, Berlin 1985.
J. Marsden and M. Me. Cracken,The Hopf Bifurcation and its Applications. Appl. Math. Sci. 19, Springer, Berlin 1976.
A. Mielke,A reduction principle for nonautonomous systems in infinite dimensional spaces. J. Diff. Eq.65, 68–88 (1986).
A. Mielke,On nonlinear problems of mixed type, a qualitative theory using infinite dimensional center manifolds. J. Dynamics and Diff. Eq., submitted.
E. Nelson,Topics in Dynamics 1, Flows. Math. Notes, Princeton University Press 1969.
A. Pazy,Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44, Springer, Berlin 1983.
C. Perron,Die Stabilitätsfrage bei Differentialgleichungen. Math. Z.32, 703–728 (1930).
M. Reed,Abstract non-linear wave equations. Lect. Notes in Math. 507, Springer, Berlin 1976.
B. Scarpellini,Stable manifold constructions for nonlinear wave equations. To appear in “Bayreuther Math. Schriften”, Heft 35 (1991).
A. Vanderbauwhede,Center manifolds, normal forms and elementary bifurcations. Dynamics Reported, Teubner, Shuttgart/Wiley, New York2, 86–169 (1989).
P. Vuillermot,Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations. Comment. Math. Helvetici64, 573–586 (1987).
A. Weinstein,Periodic nonlinear waves on a half-line. Commun. Math. Phys.99, 385–388 (1985); Erratum Ibid.107, No. 1, 177 (1986).
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Scarpellini, B. Center manifolds of infinite dimensions I: Main results and applications. Z. angew. Math. Phys. 42, 1–32 (1991). https://doi.org/10.1007/BF00962056
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DOI: https://doi.org/10.1007/BF00962056