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”Integrated semigroups” and integrated solutions to abstract Cauchy problems. (English) Zbl 0738.47037
Summary: Concepts which have been developed in the theory of weakly $\sp*$ continuous semigroups on dual Banach spaces are extended to “integrated semigroups”. The relation to integrated solutions of homogeneous and inhomogeneous abstract Cauchy problems is established. The results are illustrated for the wave equation in $L\sb 2(\bbfR\sp m)$.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations
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##### References:
 [1] Amann, H.: Parabolic evolution equations in interpolation and extrapolation spaces. J. funct. Anal. 78, 233-270 (1988) · Zbl 0654.47019 [2] Arendt, W.: Resolvent positive operators and integrated semigroups. Proc. London math. Soc. 54, 321-349 (1987) · Zbl 0617.47029 [3] Arendt, W.: Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327-352 (1987) · Zbl 0637.44001 [4] W. Arendt and H. Kellermann, Integrated solutions of Volterra integro-differential equations and applications, preprint. · Zbl 0675.45017 [5] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J. A.M; Thieme, H. R.: Perturbation theory for dual semigroups. I. the Sun-reflexive case. Math. ann. 277, 709-725 (1987) · Zbl 0634.47039 [6] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J. A.M; Thieme, H. R.: Perturbation theory for dual semigroups II. Time-dependent perturbations in the Sun-reflexive case. Proc. roy. Soc. Edinburgh sect. A 109, 145-172 (1988) · Zbl 0661.47015 [7] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J. A.M; Thieme, H. R.: Perturbation theory for dual semigroups. III. nonlinear Lipschitz continuous perturbations in the Sun-reflexive case. Pitman research notes in mathematics series 190 (1989) · Zbl 0675.47036 [8] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J. A.M; Thieme, H. R.: Perturbation theory for dual semigroups. IV. the intertwining formula and the canonical pairing. Lecture notes in pure and applied mathematics 116, 95-116 (1989) · Zbl 0699.47028 [9] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J. A.M; Thieme, H. R.: A hille-yosida theorem for a class of weakly$\ast$continuous semigroups. Semigroup forum 38, 157-178 (1989) · Zbl 0727.47025 [10] Da Prato, G.; Grisvard, P.: Maximum regularity for evolution equations by interpolation and extrapolation. J. funct. Anal. 58, 107-124 (1984) · Zbl 0593.47041 [11] G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, preprint. · Zbl 0652.34069 [12] Goldstein, J. A.: Semigroups of linear operators and applications. (1985) · Zbl 0592.47034 [13] Hille, E.; Phillips, R. S.: Functional analysis and semigroups. (1957) · Zbl 0078.10004 [14] Kellermann, H.: Integrated semigroups. Thesis (1986) [15] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., to appear. · Zbl 0604.47025 [16] Nagel, R.: Sobolev spaces and semigroups. Semesterberichte funktionalanalysis, 1-20 (1983) [17] Neubrander, F.: Integrated semigroups and their application to the abstract Cauchy problem. Pac. J. Math. 135, 111-155 (1988) · Zbl 0675.47030 [18] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023 [19] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Integr. Equations (to appear). · Zbl 0734.34059 [20] Walther, Th: Abstrakte Sobolev-räume und ihre anwendung auf störungstheorie für generatoren von C0-halbgruppen. Dissertation (1986) · Zbl 0616.46029