Rocky Mountain Journal of Mathematics

On a solution of the Cauchy problem in the weighted spaces of Beurling ultradistributions

Stevan Pilipović, Bojan Prangoski, and Daniel Velinov

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Abstract

Results of Da Prato and Sinestrari \cite {d81} on differential operators with non-dense domain but satisfying the Hille-Yosida condition, are applied in the setting of Beurling weighted spaces of ultradistributions $\DD '^{(s)}_{L^p}((0,T)\times U)$, where $U$ is open and bounded set in $\mathbb R^d$. For this purpose, the new structural theorems were given for $\DD '^{(s)}_{L^p}((0,T)\times U)$. Then a class of the Cauchy problems in the general setting of such spaces of ultradistributions is analyzed.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1937-1984.

Dates
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1457960343

Digital Object Identifier
doi:10.1216/RMJ-2015-45-6-1937

Mathematical Reviews number (MathSciNet)
MR3473163

Zentralblatt MATH identifier
1359.47037

Subjects
Primary: 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
Secondary: 35A01: Existence problems: global existence, local existence, non-existence

Keywords
Cauchy problem strict weak solution $F$-weak solution

Citation

Pilipović, Stevan; Prangoski, Bojan; Velinov, Daniel. On a solution of the Cauchy problem in the weighted spaces of Beurling ultradistributions. Rocky Mountain J. Math. 45 (2015), no. 6, 1937--1984. doi:10.1216/RMJ-2015-45-6-1937. https://projecteuclid.org/euclid.rmjm/1457960343


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References

  • W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327–352.
  • W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Springer, Basel, 2011.
  • R. Beals, On the abstract Cauchy problem, J. Funct. Anal. 10 (1972), 281–299.
  • ––––, Semigroups and abstract Gevrey spaces, J. Funct. Anal. 10 (1972), 300–308.
  • J. Chazarain, Problémes de Cauchy abstraites et applications á quelques problémes mixtes, J. Funct. Anal. 7 (1971), 386–446.
  • G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa 14 (1987), 285–344.
  • M. Hieber, Integrated semigroups and differential operators on $L^{p} $ spaces, Math. Ann. 29 (1991), 1–16.
  • H. Kellermann and M. Hieber, Inegrated semigroups, J. Funct. Anal. 84, (1989), 160–180.
  • H. Komatsu, Ultradistributions, I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo 20 (1973), 25–105.
  • H. Komatsu, Ultradistributions, II: The kernel theorem and ultradistributions with support in submanifold, J. Fac. Sci. Univ. Tokyo 24 (1977), 607–628.
  • ––––, Ultradistributions, III. Vector valued ultradistributions the theory of kernels, J. Fac. Sci. Univ. Tokyo 29 (1982), 653–718.
  • M. Kosti\' c, Generalized semigroups and cosine functions, Mathematical Institute SANU, Belgrade, 2011.
  • M. Kosti\' c and S. Pilipovi\' c, Global convoluted semigroups, Math Nachr. 280 (2007), 1727–1743.
  • ––––, Ultradistribution semigroups, Siberian Math. J. 53 (2012), 232–242.
  • P.C. Kunstmann, Distribution semigroups and abstract Cauchy problems, Trans. Amer. Math. Soc. 351 (1999), 837–856.
  • P.C. Kunstmann, Banach space valued ultradistributions and applications to abstract Cauchy problems, http://math.kit.edu/iana1/$\sim$kunstmann/media/ultra-appl.pdf.
  • R. de Laubenfels and F. Yao, Regularized semigroups of bounded semivariation, Semigroup Forum 54 (1997), 43–57.
  • A. Lunardi, Analytic semigroups and optimal regularity in parabolic problem, Birkhäuser, Basel, 1995.
  • I.V. Melnikova and A.I. Filinkov, Abstract Cauchy problems: Three approaches, Chapman & Hall/CRC, Washington, 2001.
  • R. Nagel and E. Sinestrari, Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators Dekker Lect. Notes 150 (1994), 51–70, Dekker.
  • F. Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. Math. 135 (1988), 111–155.
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
  • R. Phillips, Perturbation theory of semigroups of linear operators, Trans. Amer. Math. Soc. 74 (1953), 199–221.
  • S. Pilipović, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191–1206.
  • E. Sinestrari, On the abstract Cauchy problem of parabolic type in space of continuous functions, J. Math. Anal. Appl. 107 (1985), 16–66.
  • ––––, Hille-Yosida operators and Cauchy problems, Semigroup Forum 82 (2011), 10–34.
  • H. Schaefer, Topological vector spaces, 3rd edition, Springer-Verlag, New York, 1971.
  • L. Schwartz, Théorie des distributions I, Herman, Paris, 1966.
  • ––––, Théorie des distributions á valeurs vectorielles, I, Ann. Inst. Fourier 7 (1957), 1–141.