Advertisement

Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 349–372 | Cite as

Pseudodifferential operators with non-regular operator-valued symbols

  • Bienvenido Barraza Martínez
  • Robert DenkEmail author
  • Jairo Hernández Monzón
Article

Abstract

In this paper, we consider pseudodifferential operators with operator-valued symbols and their mapping properties, without assumptions on the underlying Banach space E. We show that, under suitable parabolicity assumptions, the \({W_p^k(\mathbb{R}^n, E)}\)-realization of the operator generates an analytic semigroup. Our approach is based on oscillatory integrals and kernel estimates for them. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. As an example, we include a discussion of coagulation–fragmentation processes.

Mathematics Subject Classification (2000)

35S05 47D06 35R20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achdou, Y., Franchi, B., Marcello, N., Tesi, M.: A qualitative model for aggregation and diffusion of \({\beta}\) -amyloid in Alzheimer’s disease. J. Math. Biol. (2012). doi: 10.1007/s00285-012-0591-0; Springer
  2. 2.
    Adam A.J.: A simplified mathematical model of tumour growth. Math. Biosci. 81, 229–244 (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Amann H.: Linear and Quasilinear Parabolic Problems, Vol. I. Birkhäuser Verlag, Basel (1995)CrossRefGoogle Scholar
  4. 4.
    Amann H.: Operator-valued fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186, 5–56 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Amann H.: Coagulation–fragmentation processes. Arch. Rot. Mech. Anal. 151, 339–366 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Amann H.: Elliptic operators with infinite dimensional state spaces. J. Evol. Equ. 1, 143–188 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Amann H., Walker C.: Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion. J. Differ. Equ. 218, 159–186 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Barraza Martínez, B.: Pseudodifferentialoperatoren mit nichtregulären banachraumwertigen Symbolen, Dissertation. Johannes Gutenberg-Universität Mainz (2009)Google Scholar
  9. 9.
    Barraza Martínez, B., Denk, R., Hernández Monzón, J.: Analytic Semigroups of Pseudodifferential Operators on Vector-Valued Sobolev Spaces. Bull. Braz. Math. Soc. (N. S.) (to appear)Google Scholar
  10. 10.
    Brzychczy S., Górniewicz L.: Continuous and discrete models of neural systems in infinite-dimensional abstract spaces. Neurocomputing 74, 2711–2715 (2011)CrossRefGoogle Scholar
  11. 11.
    Carrillo J., Desvillettes L., Fellner K.: Exponential decay towards equilibrium for the inhomogeneous Aizenman–Bak model. Commun. Math. Phys. 278, 433–451 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Clovis S., Noutchie O.: Analysis of the effects of fragmentation–coagulation in planktology. Comptes Rendus Biologie 333, 789–792 (2010)CrossRefGoogle Scholar
  13. 13.
    Denk, R., Hieber, M., Prüss, J.: \({\mathcal{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166, viii+114 (2003)Google Scholar
  14. 14.
    Denk R., Krainer T.: \({\mathcal{R}}\) -boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators. Manuscripta Math. 124, 319–342 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Filbet F.: An asymptotically stable scheme for diffusive coagulation–fragmentation models. Comm. Math. Sci. 6(2), 257–280 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Girardi M., Weis L.: Operator-valued Fourier multiplier theorems on Besov spaces. Math. Nachr. 251, 34–51 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kiehn, Chr.: Analytic Semigroup of Pseudodifferential Operators on \({L_{p} (\mathbb{R}^{n}, E)}\) , Preprint Reihe des Fachbereichs Mathematik Nr 10, Johannes Gutenberg-Universität (2001)Google Scholar
  18. 18.
    Kiehn Chr.: Analytic Semigroups of Pseudodifferential Operators on Vector-Valued Function Spaces. Shaker Verlag, Aachen (2003)zbMATHGoogle Scholar
  19. 19.
    Kumano-go H.: Pseudo-Differential Operators. MIT Press, Cambridge (1981)Google Scholar
  20. 20.
    Kunstmann, P.C., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and \({H^\infty}\) -functional calculus. In: Da Prato, et al. (ed.) Functional Analytic Methods for Evolution Equations, Lecture Notes in Math. 1855. Springer, Berlin (2004)Google Scholar
  21. 21.
    Nau, T., Saal, J.: \({\mathcal{R}}\) -sectoriality of cylindrical boundary value problems. In: Escher, J., et al. (ed.) Parabolic Problems: The Herbert Amann Festschrift, Progr. Nonlinear Differential Equations Appl. 80, pp. 479–506. Birkhuser/Springer (2011)Google Scholar
  22. 22.
    Noll A., Haller R., Heck H.: Mikhlin’s theorem for operator-valued multipliers in n variables. Math. Nach. 244, 110–130 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Portal P., Štrkalj Ž.: Pseudodifferential operators on Bochner spaces and an application. Math. Z. 253, 805–819 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Rudnicki R., Wieczorek R.: Fragmentation–coagulation models of phytoplankton. Bull. Pol. Acad. Sci. Math. 54(2), 175–191 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Walker C.: On a new model for continuous coalescence and breakage processes with diffusion. Adv. Differ. Equ. 10(2), 121–152 (2005)zbMATHGoogle Scholar
  26. 26.
    Wattis J.: An introduction to mathematical models of coagulation–fragmentation processes: a discrete deterministic mean-field approach. Physica D 222, 1–20 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Weis L.: Operator-valued Fourier multiplier theorems and maximal L p-regularity. Math. Ann. 319, 735–758 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Wrzosek D.: Weak solutions to the Cauchy problem for the diffusive discrete coagulation–fragmentation system. J. Math. Anal. Appl. 289, 405–418 (2004)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Bienvenido Barraza Martínez
    • 1
  • Robert Denk
    • 2
    Email author
  • Jairo Hernández Monzón
    • 1
  1. 1.Departamento de MatemáticasUniversidad del NorteBarranquillaColombia
  2. 2.Fachbereich MathematikUniversität KonstanzKonstanzGermany

Personalised recommendations