Skip to main content
SpringerLink
Log in

A solution to an open problem for wave equations with generalized Wentzell boundary conditions

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

In this paper we solve an open problem put forward by A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, concerning the mixed problem for wave equations with generalized Wentzell boundary conditions. As a consequence, we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized Wentzell boundary conditions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Casarino, V., Engel, K.-J., Nagel, R., Nickel, G.: A semigroup approach to boundary feedback systems. Tübinger Berichte zur Funktionalanalysis 10, 30–43 (2000/2001)

    Google Scholar 

  2. Davies, E.B.: One Parameter Semigroups. Academic Press, London, 1980

  3. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. GTM 194, Springer, Berlin, New York, 2000

  4. Fattorini, H.O.: Ordinary differential equations in linear topological spaces, I. J. Differ. Eqs. 5, 72–105 (1969)

    MATH  Google Scholar 

  5. Fattorini, H.O.: The Cauchy Problem. Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, Reading, Mass. 1983

  6. Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland Math. Studies, 108, North-Holland Publishing Co., Amsterdam, 1985

  7. Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: C 0-semigroups generated by second order differential operators with general Wentzell boundary conditions. Proc. Amer. Math. Soc. 128, 1981–1989 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: The one dimensional wave equation with Wentzell boundary condition. In: Differential Equations and Control Theory, S. Aicovici and N. Pavel, (eds), Lect. Notes in Pure Appl. Math., Vol. 225, Marcel Dekker, 2001, 139-145. Tübinger Berichte zur Funktionalanalysis, 9, 162–168 (1999/2000)

  9. Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: The heat equation with generalized Wentzell boundary condition. J. Evolution Equations, 2, 1–19 (2002)

    Google Scholar 

  10. Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Math. Monographs, Oxford Uni. Press, New York, 1985

  11. Greiner, G.: Perturbing the boundary conditions of a generator. Houston J. Math. 13, 213–229 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Hille, E.: Functional Analysis and Semi-groups. Amer. Math. Soc. Colloquium Publications, Vol. 31, Amer. Math. Soc., New York, 1948

  13. Liang, J., Nagel, R., Xiao, T.J.: Nonautonomous heat equations with generalized Wentzell boundary conditions. J. Evolution Equations. 3, 321–331 (2003)

    MATH  Google Scholar 

  14. Phillips, R.S.: A note on the abstract Cauchy problem. Proc. Nat. Acad. Sci. U.S.A. 40, 244–248 (1954)

    MathSciNet  MATH  Google Scholar 

  15. Romanoff, N.P.: On one parameter operator groups of linear transformations I. Ann. Math. 48, 216–233 (1947)

    MathSciNet  MATH  Google Scholar 

  16. Xiao, T.J., Liang, J.: The Cauchy Problem for Higher Order Abstract Differential Equations. Lecture Notes in Math., Vol. 1701, Springer, Berlin, 1998

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Ti-Jun Xiao & Jin Liang

Authors
  1. Ti-Jun Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ti-Jun Xiao.

Additional information

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, Tübingen, Germany (e-mail: tixi@fa.uni-tuebingen.de;jili@fa.uni-tuebingen.de)

The first author acknowledges support from the Alexander-von-Humboldt Foundation and from CAS and NSFC. The second author acknowledges support from the Max-Planck Society and from CAS and EMC.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiao, TJ., Liang, J. A solution to an open problem for wave equations with generalized Wentzell boundary conditions. Math. Ann. 327, 351–363 (2003). https://doi.org/10.1007/s00208-003-0457-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-003-0457-2

Keywords

Search

Navigation