Floquet problem and center manifold reduction for ordinary differential operators with periodic coefficients in Hilbert spaces. (English) Zbl 1484.34144
St. Petersbg. Math. J. 32, No. 3, 531-550 (2021) and Algebra Anal. 32, No. 3, 191-218 (2020).
Authors’ abstract: A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.
Reviewer: Adriana Buică (Cluj-Napoca)
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
Keywords:
periodic coefficients; center manifold reduction; asymptotics of solutions to differential equations; Floquet theorem; differential equations in Hilbert spacesReferences:
| [1] | Hi E. Hille and R. Phillips, Functional analysis and semi-groups, Colloquium Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. 0089373 · Zbl 0078.10004 |
| [2] | Ko V. A. Kondratiev, Boundary problems for elliptic equations in domains with conical or angular points, Tr. Moskov. Mat. Obshch. 16 (1967), 209-292; English transl., Trans. Moscow Math. Soc. 16 (1967), 227-313. 0226187 · Zbl 0194.13405 |
| [3] | KM1 V. A. Kozlov and V. G. Maz\textprime ya, Differential equations with operator coefficients with applications to boundary value problems for partial differential equations, Springer Monogr. Math., Springer-Verlag, Berlin, 1999. 1729870 · Zbl 0920.35003 |
| [4] | KM2 \bysame , An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities, J. Funct. Anal. 217 (2004), no. 2, 448-488. 2102574 · Zbl 1068.35180 |
| [5] | KMR V. A. Kozlov, V. G. Maz\textprime ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Math. Surveys Monogr., vol. 52, Amer. Math. Soc., Providence, RI, 1997. 1469972 · Zbl 0947.35004 |
| [6] | Ku81 P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4, 3-52; English transl., Russian Math. Surveys 37 (1982), no. 4, 1-60. 667973 · Zbl 0519.35003 |
| [7] | KuchBook \bysame , Floquet theory for partial differential equations, Oper. Theory Adv. Appl., vol. 60, Birkh\"auser, Basel, 1993. 1232660 · Zbl 0789.35002 |
| [8] | KuchRev \bysame , An overview of periodic elliptic operators, Bull. Amer. Math. Soc. 53 (2016), no. 3, 343-414. 3501794 · Zbl 1346.35170 |
| [9] | LM J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren Math. Wiss., Bd 181, Springer-Verlag, New York-Heidelberg, 1972. 0350177 · Zbl 0223.35039 |
| [10] | M1 A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988), no. 1, 51-66. 929221 · Zbl 0647.35034 |
| [11] | M \bysame , Hamiltonian and Lagrangian flows on center manifolds., Lecture Notes in Math., vol. 1489, Springer-Verlag, Berlin, 1991. 1165943 · Zbl 0747.58001 |
| [12] | N1 S. A. Nazarov, Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 101-112; English transl., Math. USSR-Izv. 18 (1982), no. 1, 89-98. 607578 · Zbl 0483.35032 |
| [13] | na417 \bysame , Properties of spectra of boundary value problems in cylindrical and quasi-cylindrical domains, Sobolev Spaces in Math. II, Int. Math. Ser., vol. 9, Springer, New York, 2008, pp. 261-309. 2484629 · Zbl 1208.35093 |
| [14] | NaPl S. A. Nazarov and B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expos. Math., vol. 13, Walter de Gruyter \(\&\) Co., Berlin, 1994. 1283387 · Zbl 0806.35001 |
| [15] | \endbiblist |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
