Investigations of retarded PDEs of second order in time using the method of inertial manifolds with delay.
By constructing inertial manifolds with delay (IMD) for dissipative systems of second order in time, the author studies the different asymptotic properties of solutions. The author gives the proof for the retarded case and notice that even for the nonretarded case it has not been done before in the context of systems of second order in time. Using IMD, the author constructs approximate inertial manifolds; containing all the stationary solutions. He improves the result of [\textit{I. D. Chueshov} and \textit{A. V. Rezounenko}, C. R. Acad. Sci., Paris, Sér. I 321, 607--612 (1995; Zbl 0845.35129)] on the finite number of essential modes and gives a new characterization of the \(\mathcal{K}\)-invariant manifold constructed in [\textit{M. Taboada} and \textit{Y. You}, J. Differ. Equations 114, No. 2, 337--369 (1994; Zbl 0815.34067)]. In contrast to the case of parabolic equations with delay [\textit{L. Boutet de Monvel, I. D. Chueshov} and \textit{A. V. Rezounenko}, Nonlinear Anal., Theory Methods Appl. 34, No. 6, 907--925 (1998; Zbl 0954.34064); \textit{A. V. Rezounenko}, Discrete Contin. Dyn. Syst. 6, No. 4, 829--840 (2000; Zbl 1011.37046); J. Math. Anal. Appl. 282, No. 2, 614--628 (2003; Zbl 1039.35133)], the investigations of this paper are crucially based upon a special choice of the norm introduced in [\textit{X. Mora}, Pitman Res. Notes Math. Ser. 155, 172--183 (1987; Zbl 0642.35061)].
retarded nonlinear partial differential equations
scientific article; zbMATH DE number 2162433
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