Skip to main content
Log in

The existence of a global classical solution of the initial-boundary value problem for ▭u+u3=f

  • Published:
Archive for Rational Mechanics and Analysis 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. Agmon, S., The L P approach to the Dirichlet problem. Ann. Scuola. Norm. Sup. Pisa 13, 49–92 (1959).

    Google Scholar 

  2. Browder, F. E., On non-linear wave equations. Math. Z. 80, 249–264 (1962).

    Google Scholar 

  3. Browder, F. E., & W. A. Strauss, Scattering for non-linear wave equations. Pacific J. Math. 13, 23–43 (1963).

    Google Scholar 

  4. Courant, R., & D. Hilbert, Methoden der mathematischen Physik, Vol.II. Berlin: Springer 1937.

    Google Scholar 

  5. Friedrichs, K. O., On differential operators in Hilbert space. Amer. J. Math. 61, 523–544 (1939).

    Google Scholar 

  6. Hille, E., & R. S. Phillips, Functional Analysis and Semi-Groups. Amer. Math. Soc. (1957).

  7. Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachrichten 4, 213–231 (1951).

    Google Scholar 

  8. Jörgens, K., Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77, 295–308 (1961).

    Google Scholar 

  9. Kiselev, A. A., & O. A. Ladyzhenskaya, On existence and uniqueness of the solution of the non-stationary problem for a viscous incompressible fluid. Izvestiya Akad. Nauk SSSR 21, 655–680 (1957).

    Google Scholar 

  10. Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon & Breach 1963.

    Google Scholar 

  11. Lions, J. L., Quelques résultatats d'existence dans des équations aux dérivées partielles nonlinéaires. Bull. Soc. Math. France 87, 245–273 (1959).

    Google Scholar 

  12. Lions, J. L., Equations différentielles opérationnelles et problèmes aux limites. Berlin-Göttingen-Heidelberg: Springer 1961.

    Google Scholar 

  13. Lions, J. L., Lectures given at the Mathematics Research Center, U.S. Army, Univ. of Wisc., 1963 (unpublished).

  14. Nirenberg, L., Remarks on strongly elliptic partial differential equations. Comm. Pure Applied Math. 8, 649–675 (1955).

    Google Scholar 

  15. Riesz, F., & B. Sz Nagy, Functional Analysis. New York: Ungar 1955.

    Google Scholar 

  16. Sather, J., The initial-boundary value problem for a non-linear hyperbolic equation in relativistic quantum mechanics. To appear, Journal of Mathematics and Mechanics.

  17. Segal, I. E., Non-linear semi-groups. Annals of Math. 78, 339–364 (1963).

    Google Scholar 

  18. Segal, I. E., The global Cauchy problem for a relativistic scalar field with power interaction. Bull. Soc. Math. France 91, 129–135 (1963).

    Google Scholar 

  19. Serrin, J., The Initial Value Problem for the Navier-Stokes Equations. Non Linear Problems. Madison: Univ. of Wisc. Press 1963.

    Google Scholar 

  20. Sobolev, S. L., On a theorem of functional analysis. Mat. Sbornik, N.S., 4, 471–497 (1938).

    Google Scholar 

  21. Strauss, W., Scattering for hyperbolic equations. Trans. Amer. Math. Soc. 108, 13–37 (1963).

    Google Scholar 

  22. Strauss, W., La décroissance asymptotique des solutions des équations d'onde non linéaires. C. R. Acad. Sc. Paris 256, 2749–2750 (1963).

    Google Scholar 

  23. Strauss, W., Les opérateurs d'onde pour des équations d'onde non-linéaires indépendantes du temps. C.R. Acad. Sc. Paris 256, 5045–5046 (1963).

    Google Scholar 

  24. Wilcox, C. H., Initial-boundary value problems for linear hyperbolic partial differential equations of the second order. Arch. Rational Mech. Analysis 10, 361–400 (1962).

    Google Scholar 

  25. Wilcox, C. H., Regularity theorems for solutions of initial-boundary value problems for linear hyperbolic equations of the second order. MRC Tech. Report 533, Univ. Wis., Madison 1964.

    Google Scholar 

  26. Wilcox, C. H., Uniform asymptotic estimates for wave packets in the quantum theory of scattering. J. Math. Phys. 6, 611–620 (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by J. Serrin

The research reported here was sponsored by the Mathematics Research Center, United States Army, University of Wisconsin at Madison, Wisconsin, under Contract No. DA-11-022-ORD-2059.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sather, J. The existence of a global classical solution of the initial-boundary value problem for ▭u+u3=f . Arch. Rational Mech. Anal. 22, 292–307 (1966). https://doi.org/10.1007/BF00285421

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00285421

Keywords

Navigation