zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. (English) Zbl 1126.47060
The author considers an evolution family $\cal{U} = (U(t,s))_{t\geq s\geq 0}$ on ${\Bbb{R}}_+$ and the integral equation $$ u(t)=U(t,s)u(s)+\int_s^tU(t,\xi)f(\xi)\,d\xi.$$ He characterizes the exponential dichotomy of the evolution family through the solvability of this integral equation in admissible function spaces which contain function spaces of $L_p$ type, the Lorentz spaces $L_{p,q}$, and many other function spaces occurring in interpolation theory. The paper applies the technique of choosing test functions related to integral equations. This technique allows to use the Banach isomorphism theorem applied to an abstract differential operator for obtaining explicit dichotomy estimates. Using the characterization of exponential dichotomy, the author proves the robustness of exponential dichotomy of evolution families on ${\Bbb{R}}_+$ under small perturbations.

47N20Applications of operator theory to differential and integral equations
47D06One-parameter semigroups and linear evolution equations
34D09Dichotomy, trichotomy
34G10Linear ODE in abstract spaces
35B35Stability of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
35K55Nonlinear parabolic equations
Full Text: DOI
[1] Ben-Artzi, A.; Gohberg, I.: Dichotomies of systems and invertibility of linear ordinary differential operators. Oper. theory adv. Appl. 56, 90-119 (1992) · Zbl 0766.47021
[2] Ben-Artzi, A.; Gohberg, I.; Kaashoek, M. A.: Invertibility and dichotomy of differential operators on a half-line. J. dynam. Differential equations 5, 1-36 (1993) · Zbl 0771.34011
[3] Brendle, S.; Nagel, R.: Partial functional equations with non-autonomous past. Discrete contin. Dyn. syst. 8, 953-966 (2002) · Zbl 1013.35080
[4] Calderon, A. P.: Spaces between L1 and L$\infty $ and the theorem of Marcinkiewicz. Studia math. 26, 273-299 (1996)
[5] Coppel, W. A.: Dichotomy in stability theory. (1978) · Zbl 0376.34001
[6] Daleckii, J. L.; Krein, M. G.: Stability of solutions of differential equations in Banach spaces. Transl. amer. Math. soc. (1974)
[7] Datko, R.: Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. anal. 3, 428-445 (1972) · Zbl 0241.34071
[8] Engel, K. J.; Nagel, R.: One-parameter semigroups for linear evolution equations. Grad texts in math. 194 (2000) · Zbl 0952.47036
[9] Huy, N. T.; Minh, N. V.: Exponential dichotomy of difference equations and applications to evolution equations on the half-line. Comput. math. Appl. 42, 301-311 (2001) · Zbl 1016.39007
[10] Huy, N. T.: Resolvents of operators and partial functional differential equations with non-autonomous past. J. math. Anal. appl. 289, 301-316 (2004) · Zbl 1063.47033
[11] Huy, N. T.: Exponentially dichotomous operators and exponential dichotomy of evolution equations on the half-line. Integral equations operator theory 48, 497-510 (2004) · Zbl 1058.34072
[12] Kato, T.: Perturbation theory for linear operators. (1980) · Zbl 0435.47001
[13] Levitan, B. M.; Zhikov, V. V.: Almost periodic functions and differential equations. (1978) · Zbl 0414.43008
[14] Lindenstrauss, J.; Tzafriri, L.: Classical Banach spaces II, function spaces. (1979) · Zbl 0403.46022
[15] Massera, J. J.; Schäffer, J. J.: Linear differential equations and function spaces. (1966) · Zbl 0243.34107
[16] Minh, N. V.; Huy, N. T.: Characterizations of dichotomies of evolution equations on the half-line. J. math. Anal. appl. 261, 28-44 (2001) · Zbl 0995.34038
[17] Minh, N. V.; Räbiger, F.; Schnaubelt, R.: Exponential dichotomy exponential expansiveness and exponential dichotomy of evolution equations on the half-line. Integral equations operator theory 32, 332-353 (1998) · Zbl 0977.34056
[18] Nagel, R.; Nickel, G.: Well-posedness for non-autonomous abstract Cauchy problems. Progr. nonlinear differential equations appl. 50, 279-293 (2002) · Zbl 1058.34074
[19] Van Neerven, J.: The asymptotic behaviour of semigroups of linear operator. Oper. theory adv. Appl. 88 (1996)
[20] G. Nickel, On evolution semigroups and wellposedness of non-autonomous Cauchy problems, PhD thesis, Tübingen, 1996 · Zbl 0880.47024
[21] Pazy, A.: Semigroup of linear operators and application to partial differential equations. (1983) · Zbl 0516.47023
[22] Perron, O.: Die stabilitätsfrage bei differentialgleichungen. Math. Z. 32, 703-728 (1930) · Zbl 56.1040.01
[23] Räbiger, F.; Schnaubelt, R.: Absorption evolution families with applications to non-autonomous diffusion processes. Tübinger ber. Funktionalanalysis 5, 335-354 (1995/1996)
[24] 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
[25] R. Schnaubelt, Exponential bounds and hyperbolicity of evolution families, PhD thesis, Tübingen, 1996 · Zbl 0880.47025
[26] Schnaubelt, R.: Exponential dichotomy of non-autonomous evolution equations. (1999) · Zbl 0936.34038
[27] Schnaubelt, R.: Asymptotically autonomous parabolic evolution equations. J. evol. Equ. 1, 19-37 (2001) · Zbl 1098.34551
[28] Triebel, H.: Interpolation theory, function spaces, differential operators. (1978) · Zbl 0387.46033