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Transactions of the American Mathematical Society

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Distribution semigroups and
abstract Cauchy problems


Author: Peer Christian Kunstmann
Journal: Trans. Amer. Math. Soc. 351 (1999), 837-856
MSC (1991): Primary 47D03, 34G10, 47A10, 46F10
DOI: https://doi.org/10.1090/S0002-9947-99-02035-8
MathSciNet review: 1443882
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Abstract | References | Similar Articles | Additional Information

Abstract: We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions are equivalent: (a) $A$ generates a distribution semigroup; (b) the convolution operator $\delta'\otimes I-\delta\otimes A$ has a fundamental solution in ${\mathcal D}'(L(E,D))$ where $D$ denotes the domain of $A$ supplied with the graph norm and $I$ denotes the inclusion $D\to E$; (c) $A$ generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.


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  • 1. W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352. MR 89a:47064
  • 2. W. Arendt, O. El-Mennaoui, V. Keyantuo, Local Integrated Semigroups: Evolution with Jumps of Regularity, J. Math. Analysis Appl. 186 (1994), 572-595. MR 95f:47065
  • 3. M. Balabane, Puissances fractionnaires d'un opérateur générateur d'un semi-groupe distribution régulier, Ann. Inst. Fourier, Grenoble, 26 (1976), 157-203. MR 53:6353
  • 4. R. Beals, On the abstract Cauchy problem, J. Funct. Analysis 10 (1972), 281-299. MR 51:8859
  • 5. R. Beals, Semigroups and abstract Gevrey spaces, J. Funct. Analysis 10 (1972), 300-308. MR 50:14355
  • 6. J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, J. Funct. Analysis 7 (1971), 386-446. MR 43:2570
  • 7. G. Da Prato, E. Sinestrari, Differential Operators with Non-dense Domain, Ann. Scuola Normale Pisa 14 (1987), 285-344. MR 89f:47062
  • 8. E. B. Davies, M. M. H. Pang, The Cauchy Problem and a Generalization of the Hille-Yosida Theorem, Proc. London Math. Soc. 55 (1987), 181-208. MR 88e:34100
  • 9. R. deLaubenfels, $C$-existence families and improperly posed problems, Semesterbericht Funktionalanalysis, Tübingen WS 1989/90, 155-171.
  • 10. R. deLaubenfels, "Existence Families, Functional Calculi and Evolution Equations," Lect. Notes in Math. 1570, Springer, 1994. MR 96b:47047
  • 11. R. deLaubenfels, Automatic extension of functional calculi, Stud. Math. 114 (1995), 237-259. MR 96f:47029
  • 12. H. O. Fattorini, Some remarks on convolution equations for vector valued distributions, Pacific J. Math. 66 (1976), 347-371. MR 57:13471
  • 13. H. O. Fattorini, Vector valued distributions having a smooth convolution inverse, Pacific J. Math. 90 (1980), 347-372. MR 82f:46046
  • 14. H. O. Fattorini, The Cauchy problem, Addison-Wesley, 1983. MR 84g:34003
  • 15. V. Keyantuo, Semi-groupes distributions, semi-groupes intrégrés et problèmes d'évolution, Thèse, Université de Franche-Comté, Besançon 1992.
  • 16. J. L. Lions, Les semi-groupes distributions, Portugal. Math. 19 (1960), 141-164. MR 26:611
  • 17. Yu. I. Lyubich, Investigation of the deficiency of the abstract Cauchy problem, Soviet Math. Dokl. 7 (1960), 166-169.
  • 18. I. Miyadera, M. Okubo, N. Tanaka, On Integrated Semigroups which are not Exponentially Bounded, Proc. Japan Acad., Ser. A 69 (1993), 199-204. MR 95a:47037
  • 19. L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. MR 35:730
  • 20. L. Schwartz, Théorie des distributions à valeurs vectorielles, I: Ann. Inst. Fourier 7 (1957), 1-141; II: Ann. Inst. Fourier 8 (1958), 1-209. MR 21:6534; MR 22:8322
  • 21. R. Shiraishi, Y. Hirata, Convolution Maps and Semi-group Distributions, J. Sci. Hiroshima Univ. Ser. A-I 28 (1964), 71-88. MR 30:1396
  • 22. N. Tanaka, I. Miyadera, $C$-Semigroups and the abstract Cauchy problem, J. Math. Analysis Appl. 170 (1992), 196-206. MR 93j:47061
  • 23. N. Tanaka, N. Okazawa, Local $C$-semigroups and local integrated semigroups, Proc. London Math. Soc. 61 (1990), 63-90. MR 91b:47093
  • 24. H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Analysis Appl. 152 (1990), 416-447. MR 91k:47093
  • 25. S. Wang, A kind of smooth distribution semigroups and integrated semigroups, J. Funct. Analysis, to appear.

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Additional Information

Peer Christian Kunstmann
Affiliation: Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel, Ludewig- Meyn-Straße 4, D-24098 Kiel, Germany
Address at time of publication: Mathematisches Institut I der Universität Karlsruhe, Englerstraße 2, D-76128 Karlsruhe, Germany
Email: peer.kunstmann@math.uni-karlsruhe.de

DOI: https://doi.org/10.1090/S0002-9947-99-02035-8
Received by editor(s): October 17, 1995
Received by editor(s) in revised form: February 6, 1997
Article copyright: © Copyright 1999 American Mathematical Society

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