Abstract and Applied Analysis

On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

Alberto Favaron and Angelo Favini

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Abstract

For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 275494, 37 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449740

Digital Object Identifier
doi:10.1155/2013/275494

Mathematical Reviews number (MathSciNet)
MR3139482

Zentralblatt MATH identifier
1310.47058

Citation

Favaron, Alberto; Favini, Angelo. On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 275494, 37 pages. doi:10.1155/2013/275494. https://projecteuclid.org/euclid.aaa/1393449740


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References

  • A. Favini and A. Yagi, “Multivalued linear operators and degenerate evolution equations,” Annali di Matematica Pura ed Applicata, vol. 163, pp. 353–384, 1993.
  • A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, vol. 215, Marcel Dekker, New York, NY, USA, 1999.
  • A. Yagi, “Generation theorem of semigroup for multivalued linear operators,” Osaka Journal of Mathematics, vol. 28, no. 2, pp. 385–410, 1991.
  • P. Acquistapace and B. Terreni, “Existence and sharp regularity results for linear parabolic nonautonomous integro-differential equations,” Israel Journal of Mathematics, vol. 53, no. 3, pp. 257–303, 1986.
  • G. Da Prato, M. Iannelli, and E. Sinestrari, “Regularity of solutions of a class of linear integro-differential equations in Banach spaces,” Journal of Integral Equations, vol. 8, no. 1, pp. 27–40, 1985.
  • G. Da Prato and M. Iannelli, “Existence and regularity for a class of integro-differential equations of parabolic type,” Journal of Mathematical Analysis and Applications, vol. 112, no. 1, pp. 36–55, 1985.
  • G. Da Prato, “Abstract differential equations, maximal regularity and linearization,” in Proceedings of the Symposia in Pure Mathematics, vol. 45, part 1, pp. 359–370, 1986.
  • A. Lunardi and E. Sinestrari, “${C}^{\alpha }$-regularity for nonautonomous linear integro-differential equations of parabolic type,” Journal of Differential Equations, vol. 63, no. 1, pp. 88–116, 1986.
  • A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, Switzerland, 1995.
  • J. Prüss, “On linear Volterra equations of parabolic type in Banach spaces,” Transactions of the American Mathematical Society, vol. 301, no. 2, pp. 691–721, 1987.
  • E. Sinestrari, “On the abstract Cauchy problem of parabolic type in spaces of continuous functions,” Journal of Mathematical Analysis and Applications, vol. 107, no. 1, pp. 16–66, 1985.
  • E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, USA, 1957.
  • C. Wild, “Semi-groupes de croissance $\alpha <1$ holomorphes,” Comptes Rendus de l'Académie des Sciences, vol. 285, no. 6, pp. A437–A440, 1977 (French).
  • K. Taira, “On a degenerate oblique derivative problem with interior boundary conditions,” Proceedings of the Japan Academy, vol. 52, no. 9, pp. 484–487, 1976.
  • K. Taira, “The theory of semigroups with weak singularity and its applications to partial differential equations,” Tsukuba Journal of Mathematics, vol. 13, no. 2, pp. 513–562, 1989.
  • W. von Wahl, “Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen Höldersteiger Funktionen,” Nachrichten der Akademie der Wissenschaften in Göttingen. II, vol. 11, pp. 231–258, 1972 (German).
  • W. von Wahl, “Lineare und semilineare parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 43, pp. 234–262, 1975 (German).
  • W. von Wahl, “Neue Resolventenabschätzungen für elliptische Differentialoperatoren und semilineare parabolische Gleichungen,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 46, pp. 179–204, 1977 (German).
  • A. Favaron, “Optimal time and space regularity for solutions of degenerate differential equations,” Central European Journal of Mathematics, vol. 7, no. 2, pp. 249–271, 2009.
  • A. Favaron and A. Favini, “Maximal time regularity for degenerate evolution integro-differential equations,” Journal of Evolution Equations, vol. 10, no. 2, pp. 377–412, 2010.
  • A. Favini and A. Yagi, “Space and time regularity for degenerate evolution equations,” Journal of the Mathematical Society of Japan, vol. 44, no. 2, pp. 331–350, 1992.
  • A. Favini, A. Lorenzi, and H. Tanabe, “Singular integro-differential equations of parabolic type,” Advances in Differential Equations, vol. 7, no. 7, pp. 769–798, 2002.
  • A. Favini, A. Lorenzi, and H. Tanabe, “Singular evolution integro-differential equations with kernels defined on bounded intervals,” Applicable Analysis, vol. 84, no. 5, pp. 463–497, 2005.
  • A. Favaron and A. Favini, “Fractional powers and interpolation theory for multivalued linear operators and applications to degenerate differential equations,” Tsukuba Journal of Mathematics, vol. 35, no. 2, pp. 259–323, 2011.
  • R. Cross, Multivalued Linear Operators, vol. 213, Marcel Dekker, New York, NY, USA, 1998.
  • A. Favini and A. Yagi, “Quasilinear degenerate evolution equations in Banach spaces,” Journal of Evolution Equations, vol. 4, no. 3, pp. 421–449, 2004.
  • H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland Publishing, Amsterdam, The Netherlands, 1978.
  • H. Berens, Interpolationsmethoden zur Behandlung von Approximationsprozessen auf Banachräumen, Lecture Notes in Mathematics, Springer, Berlin, Germany, 1968 (German).
  • E. Gagliardo, “Interpolation d'espaces de Banach et applications. III,” Comptes Rendus de l'Académie des Sciences, vol. 248, pp. 3517–3518, 1959 (French).
  • P. Grisvard, “Commutativité de deux foncteurs d'interpolation et applications,” Journal de Mathématiques Pures et Appliquées, vol. 45, pp. 143–206, 1966 (French).
  • H. Berens and P. L. Butzer, “Approximation theorems for semi-group operators in intermediate spaces,” Bulletin of the American Mathematical Society, vol. 70, pp. 689–692, 1964.
  • A. Belleni-Morante, “An integro-differential equation arising from the theory of heat conduction in rigid materials with memory,” Unione Matematica Italiana, vol. 15, no. 2, pp. 470–482, 1978.
  • G. Da Prato and M. Iannelli, “Linear integro-differential equations in Banach spaces,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 62, pp. 207–219, 1980.
  • G. Da Prato, M. Iannelli, and E. Sinestrari, “Temporal regularity for a class of integro-differential equations in Banach spaces,” Bollettino Unione Matematica Italiana, vol. 2, no. 1, pp. 171–185, 1983.
  • A. Favini, A. Lorenzi, and H. Tanabe, “Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: ${L}^{2}$-theory,” Journal of the Mathematical Society of Japan, vol. 61, no. 1, pp. 133–176, 2009.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 2001, Reprint of the 1998 edition.
  • B. D. Coleman and M. E. Gurtin, “Equipresence and constitutive equations for rigid heat conductors,” Zeitschrift für Angewandte Mathematik und Physik, vol. 18, pp. 199–208, 1967.
  • M. E. Gurtin, “On the thermodynamics of materials with memory,” Archive for Rational Mechanics and Analysis, vol. 28, no. 1, pp. 40–50, 1968.
  • R. K. Miller, “An integro-differential equation for rigid heat conductors with memory,” Journal of Mathematical Analysis and Applications, vol. 66, no. 2, pp. 313–332, 1978.
  • J. W. Nunziato, “On heat conduction in materials with memory,” Quarterly of Applied Mathematics, vol. 29, pp. 187–204, 1971.
  • A. Favini, A. Lorenzi, H. Tanabe, and A. Yagi, “An ${L}^{p}$-approach to singular linear parabolic equations in bounded domains,” Osaka Journal of Mathematics, vol. 42, no. 2, pp. 385–406, 2005.