References
Agmon, S., The L P approach to the Dirichlet problem. Ann. Scuola. Norm. Sup. Pisa 13, 49–92 (1959).
Browder, F. E., On non-linear wave equations. Math. Z. 80, 249–264 (1962).
Browder, F. E., & W. A. Strauss, Scattering for non-linear wave equations. Pacific J. Math. 13, 23–43 (1963).
Courant, R., & D. Hilbert, Methoden der mathematischen Physik, Vol.II. Berlin: Springer 1937.
Friedrichs, K. O., On differential operators in Hilbert space. Amer. J. Math. 61, 523–544 (1939).
Hille, E., & R. S. Phillips, Functional Analysis and Semi-Groups. Amer. Math. Soc. (1957).
Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachrichten 4, 213–231 (1951).
Jörgens, K., Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77, 295–308 (1961).
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).
Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon & Breach 1963.
Lions, J. L., Quelques résultatats d'existence dans des équations aux dérivées partielles nonlinéaires. Bull. Soc. Math. France 87, 245–273 (1959).
Lions, J. L., Equations différentielles opérationnelles et problèmes aux limites. Berlin-Göttingen-Heidelberg: Springer 1961.
Lions, J. L., Lectures given at the Mathematics Research Center, U.S. Army, Univ. of Wisc., 1963 (unpublished).
Nirenberg, L., Remarks on strongly elliptic partial differential equations. Comm. Pure Applied Math. 8, 649–675 (1955).
Riesz, F., & B. Sz Nagy, Functional Analysis. New York: Ungar 1955.
Sather, J., The initial-boundary value problem for a non-linear hyperbolic equation in relativistic quantum mechanics. To appear, Journal of Mathematics and Mechanics.
Segal, I. E., Non-linear semi-groups. Annals of Math. 78, 339–364 (1963).
Segal, I. E., The global Cauchy problem for a relativistic scalar field with power interaction. Bull. Soc. Math. France 91, 129–135 (1963).
Serrin, J., The Initial Value Problem for the Navier-Stokes Equations. Non Linear Problems. Madison: Univ. of Wisc. Press 1963.
Sobolev, S. L., On a theorem of functional analysis. Mat. Sbornik, N.S., 4, 471–497 (1938).
Strauss, W., Scattering for hyperbolic equations. Trans. Amer. Math. Soc. 108, 13–37 (1963).
Strauss, W., La décroissance asymptotique des solutions des équations d'onde non linéaires. C. R. Acad. Sc. Paris 256, 2749–2750 (1963).
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).
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).
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.
Wilcox, C. H., Uniform asymptotic estimates for wave packets in the quantum theory of scattering. J. Math. Phys. 6, 611–620 (1965).
Author information
Authors and Affiliations
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
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00285421