Skip to main content
Log in

Positivity and regularity of hyperbolic Volterra equations in Banach spaces

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Carr, R.W., Hannsgen, K.B.: A nonhomogeneous integrodifferential equation in Hilbert space. SIAM J. Math. Anal.10, 961–984 (1979)

    Google Scholar 

  2. Carr, R.W., Hannsgen, K.B.: Resolvent formulas for a Volterra equation in Hilbert space. SIAM J. Math. Anal.13, 453–483 (1982)

    Google Scholar 

  3. Clement, P., Nohel, J.A.: Abstract linear and nonlinear Volterra equations preserving positivity. SIAM J. Math. Anal.10, 365–388 (1979)

    Google Scholar 

  4. Dafermos, C.R.: Asymptotic stability in viscoelasticity. Arch. Rat. Mech. Anal.37, 297–308 (1970)

    Google Scholar 

  5. Dafermos, C.R.: An abstract Volterra equation with applications to linear viscoelasticity. J. Differ. Equations7, 554–469 (1970)

    Google Scholar 

  6. Da Prato, G., Iannelli, M.: Linear integro-differential equations in Banach space. Rend. Sem. Math. Padova62, 207–219 (1980)

    Google Scholar 

  7. Desch, W., Grimmer, R.: Propagation of singularities for integrodifferential equations (to appear)

  8. Desch, W., Grimmer, R.: Smoothing properties of linear Volterra integrodifferential equations. Preprint

  9. Desch, W., Grimmer, R., Schappacher, W.: Some considerations for linear integrodifferential equations. J. Math. Anal. Appl.104, 219–234 (1984)

    Google Scholar 

  10. Greiner, G., Voigt, J., Wolff, M.: On the spectral bound of the generator of a semigroup of positive operators. J. Oper. Theory5, 245–256 (1981)

    Google Scholar 

  11. Grimmer, R., Zeman, M.: Wave propagation for linear integrodifferential equations in Banach space. J. Differ. Equations54, 274–282 (1984)

    Google Scholar 

  12. Hannsgen, K.B., Wheeler, R.L.: Behavior of the solution of a Volterra equation as a parameter tends to infinity. J. Integral Equations7, 229–237 (1984)

    Google Scholar 

  13. Hrusa, W.J., Renardy, M.: On wave propagation in linear viscoelasticity. Quart. Appl. Math.43, 237–253 (1985)

    Google Scholar 

  14. Joseph, D.D., Narain, A., Riccius, O.: Shear wave speeds and elastic moduli for different liquids. Preprint.

  15. Miller, R.K.: Nonlinear Volterra integral equations. Menlo Park: Benjamin 1971

    Google Scholar 

  16. Miller, R.K., Wheeler, R.L.: Asymptotic behavior for a linear Volterra integral equation in Hilbert space. J. Differ. Equations23, 270–284 (1977)

    Google Scholar 

  17. Noren, R.D.: UniformL 1-behavior of the solution of a Volterra equation with a parameter. Preprint

  18. Pipkin, A.C.: Lectures on viscoelasticity theory. Appl. Math. Sci.7. Berlin, Heidelberg, New York: Springer 1972

    Google Scholar 

  19. Prüss, J.: Lineare Volterra Gleichungen in Banach-Räumen. Habilitationsschrift, Paderborn (1984)

  20. Prüss, J.: On linear Volterra equations of parabolic type in Banach spaces. Trans. Am. Math. Soc.301, 691–721 (1987)

    Google Scholar 

  21. Prüss, J.: Bounded solutions of Volterra equations SIAM J. Math. Anal. (to appear)

  22. Renardy, M.: Some remarks on the propagation and non-propagation of discontinuities in linearly viscoelastic liquids. Rheol. Acta21, 251–254 (1982)

    Google Scholar 

  23. Travis, C.C., Webb, G.F.: Second order differential equations in Banach space. In: Nonlinear equations in abstract spaces. Ed. V. Lakshmikantham. London, New York: Academic Press 1978

    Google Scholar 

  24. Widder, D.V.: The Laplace transform. Princeton: Princeton University Press 1941

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This research was done while the author was visiting at the Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24060, USA

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prüss, J. Positivity and regularity of hyperbolic Volterra equations in Banach spaces. Math. Ann. 279, 317–344 (1987). https://doi.org/10.1007/BF01461726

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01461726

Keywords

Navigation