# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Invariant manifolds of admissible classes for semi-linear evolution equations. (English) Zbl 1178.34070
This paper is a nice contribution to the study of asymptotic behavior of solutions of semilinear evolution equations. Using the characterization of the exponential dichotomy of evolution equations, the existence of invariant manifolds for the evolution equation is obtained under more general conditions on the nonlinear term $f$. And, the assumption of uniform Lipschitz continuity of the nonlinear term $f$ is actually replaced by its non-uniform Lipschitz continuity. Generally speaking, this is a good article being worth reading.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34C45 Invariant manifolds (ODE)
Full Text:
##### References:
 [1] Aulbach, B.; Minh, N. V.: Nonlinear semigroups and the existence and stability of semilinear nonautonomous evolution equations, Abstr. appl. Anal. 1, 351-380 (1996) · Zbl 0934.34051 · doi:10.1155/S108533759600019X · http://www.hindawi.com/journals/aaa/volume-1/issue-4.html [2] Bates, P.; Jones, C.: Invariant manifolds for semilinear partial differential equations, Dyn. rep. 2, 1-38 (1989) · Zbl 0674.58024 [3] Calderon, A. P.: Spaces between L1 and L$\infty$ and the theorem of Marcinkiewicz, Studia math. 26, 273-299 (1996) [4] Daleckii, Ju.L.; Krein, M. G.: Stability of solutions of differential equations in Banach spaces, (1974) [5] Engel, K. J.; Nagel, R.: One-parameter semigroups for linear evolution equations, Grad texts in math. 194 (2000) · Zbl 0952.47036 [6] Hale, J.; Magalhães, L. T.; Oliva, W. M.: Dynamics in infinite dimensions, Appl. math. Sci. 47 (2002) · Zbl 1002.37002 [7] Henry, D.: Geometric theory of semilinear parabolic equations, Lecture notes in math. 840 (1981) · Zbl 0456.35001 [8] Hirsch, N.; Pugh, C.; Shub, M.: Invariant manifolds, Lecture notes in math. 183 (1977) · Zbl 0355.58009 [9] Huy, Nguyen Thieu: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. funct. Anal. 235, 330-354 (2006) · Zbl 1126.47060 · doi:10.1016/j.jfa.2005.11.002 [10] Lindenstrauss, J.; Tzafriri, L.: Classical Banach spaces II, function spaces, (1979) · Zbl 0403.46022 [11] Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems, (1995) · Zbl 0816.35001 [12] Martin, R. H.: Nonlinear operators and differential equations in Banach spaces, (1976) · Zbl 0333.47023 [13] Massera, J. J.; Schäffer, J. J.: Linear differential equations and function spaces, (1966) · Zbl 0243.34107 [14] Minh, N. V.; Räbiger, F.; Schnaubelt, R.: Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral equations operator theory 32, 332-353 (1998) · Zbl 0977.34056 · doi:10.1007/BF01203774 [15] Minh, N. V.; Wu, J.: Invariant manifolds of partial functional differential equations, J. differential equations 198, 381-421 (2004) · Zbl 1061.34056 · doi:10.1016/j.jde.2003.10.006 [16] Nagel, R.; Nickel, G.: Well-posedness for non-autonomous abstract Cauchy problems, Progr. nonlinear differential equations appl. 50, 279-293 (2002) · Zbl 1058.34074 [17] G. Nickel, On evolution semigroups and wellposedness of non-autonomous Cauchy problems, PhD thesis, Tübingen, 1996 · Zbl 0880.47024 [18] Nitecki, Z.: An introduction to the orbit structure of diffeomorphisms, (1971) · Zbl 0246.58012 [19] Pazy, A.: Semigroup of linear operators and application to partial differential equations, (1983) · Zbl 0516.47023 [20] Räbiger, F.; Schnaubelt, R.: The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup forum 48, 225-239 (1996) · Zbl 0897.47037 · doi:10.1007/BF02574098 [21] R. Schnaubelt, Exponential bounds and hyperbolicity of evolution families, PhD thesis, Tübingen, 1996 · Zbl 0880.47025 [22] R. Schnaubelt, Exponential dichotomy of non-autonomous evolution equations, Habilitationsschrift, Tübingen, 1999 · Zbl 0936.34038 [23] Sell, G. R.; You, Y.: Dynamics of evolutionary equations, Appl. math. Sci. 143 (2002) · Zbl 1254.37002 [24] Triebel, H.: Interpolation theory, function spaces, differential operators, (1978) · Zbl 0387.46033