Abstract
In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of the algebraic structure (for the usual convolution product *) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of k-distribution semigroups to extend previous concepts of distribution semigroups.
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Arendt, W., Kellermann, H.: Integrated solutions of Volterra integrodifferential equations and applications. In: Volterra Integrodifferential Equations in Banach spaces and Applications, Trento, 1987, Pitman Res. Notes Math. Ser., vol. 190, pp. 21–51. Longman sci. Tech., Harlow (1989)
Arendt W., El-Mennaoui O., Keyantuo V.: Local integrated semigroups: evolution with jumps of regularity. J. Math. Anal. Appl. 186(2), 572–595 (1994)
Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)
Bäumer, B.: Approximate solutions to the abstract Cauchy problem. In: Evolution Equations and their Application in Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, Vol. 215, pp. 33–41. Marcel Dekker, New York (2001)
Balabane M., Emamirad H.A.: L p estimates for Schrödinger evolution equations. Trans. Am. Math. Soc. 292(1), 357–373 (1985)
Carron G., Coulhon T., Ouhabaz E.-M.: Gaussian estimates and L p-boundedness of Riesz means. J. Evol. Equ. 2(3), 299–317 (2002)
van Casteren, J.A.: “Generators of Strongly Continuous Semigroups”. Research Notes in Mathematics, Vol. 115. Pitman Advanced Publishing Program, Boston (1985)
Chazarain J.: Problèmes de Cauchy abstraits et applications à à quelques problèmes mixtes. J. Funct. Anal. 7, 386–446 (1971)
Ciorănescu, I.: Local convoluted semigroups, in: Evolution Equations (Baton Rouge, LA, 1992), pp. 107–122. Dekker, New York (1995)
Ciorănescu I., Lumer G.: Problèmes d’évolution régularisés par un noyau général K(t). Formule de Duhamel, prolongements, théorèmes de génération. C. R. Acad. Sci. Paris Sér. I Math. 319(12), 1273–1278 (1994)
Dubois R.M., Lumer G.: Formule de Duhamel abstraite. Arch. Math. 43(1), 49–56 (1984)
Duhamel, J.M.C.: Mémoire sur la méthode générale relative au mouvement de la chaleur dans les corps solides plongés dans les milieux dont la température varie avec le temps. J. Ec. Polyt. Paris 14, Cah. 22, 20 (1833)
Fattorini H.O.: The Cauchy Problem. Addison-Wesley Publishing Co., Reading (1983)
Hieber M.: Integrated semigroups and differential operators on \({L^p(\mathbb{R}^N)}\) . Math. Ann. 291(1), 1–16 (1991)
Kaiser C., Weis L.: Perturbation theorems for α-times integrated semigroups. Arch. Math. 81(1), 215–228 (2003)
Keyantuo V.: Integrated semigroups and related partial differential equations. J. Math. Anal. Appl. 212(1), 135–153 (1997)
Keyantuo V., Müller C., Vieten P.: The Hille–Yosida theorem for local convoluted semigroups. Proc. Edinb. Math. Soc. 46(2), 357–372 (2003)
Keyantuo V., Lizama C., Miana P.: Algebra homomorphisms defined via convoluted semigroups and cosine functions. J. Funct. Anal. 257(11), 3454–3487 (2009)
Kostić M.: Generalized Semigroups and Cosine Functions. Mathematical Institute, Belgrade (2011)
Kostić M., Pilipović S.: Global convoluted semigroups. Math. Nachr. 280(15), 1727–1743 (2007)
Kunstmann P.C.: Distribution semigroups and abstract Cauchy problems. Trans. Am. Math. Soc 351(2), 837–856 (1999)
Kunstmann P.C., Mijatovic M., Pilipovic S.: Classes of distribution semigroups. Studia Math. 187(1), 37–58 (2008)
Kuo C.C., Shaw S.Y.: On α-times integrated C-semigroups and the abstract Cauchy problem. Studia Math. 142(3), 201–217 (2000)
Li Y-C., Shaw S-Y.: On local α-times integrated C-semigroups. Abst. Appl. Anal. 207, 1–18 (2007). doi:10.1155/2007/34890
Lions J.L.: Les semi-groupes distributions. Portugalia Math. 19, 141–164 (1960)
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, Vol. III. Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer, New York (1973)
Melnikova I., Filinkov A.: Abstract Cauchy Problems: Three Approaches. Chapman-Hall/CRC, New York (2001)
Miana P.J.: α-Times integrated semigroups and fractional derivation. Forum Math. 14(1), 23–46 (2002)
Miana P.J.: Local and global solutions of well-posed integrated Cauchy problems. Studia Math. 187(3), 219–232 (2008)
Miana, P.J., Poblete, V.: Sharp extensions for convoluted solutions of wave equations (2013, Preprint)
Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon-Breach, New York (1993)
Seeley R.T.: Extension of \({C^\infty}\) functions defined in a half space. Proc. Am. Math. Soc. 15, 625–626 (1964)
Tanaka N., Okazawa N.: Local C-semigroups and local integrated semigroups. Proc. Lond. Math. Soc. (3) 61(1), 63–90 (1990)
Umarov S.: On fractional Duhamel’s principle and its applications. J. Differ. Equ. 252(10), 5217–5234 (2012)
Wang S.W.: Quasi-distribution semigroups and integrated semigroups. J. Funct. Anal. 146(2), 352–381 (1997)
Wang S.W., Gao M.C.: Automatic extensions of local regularized semigroups and local regularized cosine funtions. Proc. Am. Math. Soc. 127(6), 1651–1663 (1999)
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Keyantuo, V., Miana, P.J. & Sánchez-Lajusticia, L. Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems. Integr. Equ. Oper. Theory 77, 211–241 (2013). https://doi.org/10.1007/s00020-013-2076-y
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DOI: https://doi.org/10.1007/s00020-013-2076-y