Abstract
There is given a revision of the formulation and the proof of the theorem regarding the global unique solvability in the class of weak (energy) solutions of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n ⩾ 3) without boundary (see the author's previous paper in Zap. Nauchn. Sem. LOMl, Vol. 182, 1990). Under natural additional assumptions it is proved that if the initial data u(0, x), ∂tu(0, x) are smoother:
then also the weak solution is smoother:
.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).
Ph. Brenner, “On Lp-Lp′, estimates for the wave-equation,” Math. Z.,145, No. 3, 251–254 (1975).
Ph. Brenner, “Lp-Lp′ estimates for Fourier integral operators related to hyperbolic equations,” Math. Z.,152, 273–286 (1977).
Ph. Brenner, “On the existence of global smooth solutions of certain semi-linear hyperbolic equations,” Math. Z.,167, 99–135 (1979).
Ph. Brenner and W. von Wahl, “Global classical solutions of nonlinear wave equations,” Math. Z.,176, No. 2, 87–121 (1981).
Ph. Brenner, “On space — time means and strong global solutions of nonlinear hyperbolic equations,” Math. Z.,201, No. 1, 45–55 (1989).
J. Ginibre and G. Velo, “The global Cauchy problem for the nonlinear Klein-Gordon equation,” Math. Z.,189, No. 4, 487–505 (1985).
J. Ginibre and G. Velo, “The global Cauchy problem for the nonlinear Klein-Gordon equation. II,” Ann. Inst. H. Poincaré Anal. Non Linéaire,6, No. 1, 15–35 (1989).
E. Heinz and W. von Wahl, “Zu einem Satz von F. E. Browder über nichlineare Wellengleichungen,” Math. Z.,141, 33–45 (1975).
K. Jörgens, “Das Anfangswertproblem im Grossen für eine Klasse nichlinearer Wellengleichungen,” Math. Z.,77, 295–308 (1961).
L. V. Kapitanskii, “Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,171, 106–162 (1989).
L. V. Kapitanskii, “Some generalizations of the Strichartz-Brenner inequality,” Algebra i Analiz,1, No. 3, 127–159 (1989).
L. V. Kapitanskii, “The Cauchy problem for a semilinear wave equation. II,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,182, 38–85 (1990).
L. V. Kapitanskii, “The Cauchy problem for a semilinear wave equation,” in: Proc. Fourth Conf. on Differential Equations and Their Applications (Rousse, Bulgaria, August 13–19, 1989).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris (1969).
T. Motai, “Existence of global strong solutions for nonlinear Klein-Gordon equation,” Funkcial. Ekvac.,32, No. 1, 125–144 (1989).
H. Pecher, LP-AbschÄtzungen und klassische Lösungen für nichtlineare Wellengleichungen. I,” Math. Z.,150, No. 2, 159–183 (1976).
H. Pecher, LP-AbschÄtzungen und klassische Lösungen für nichtlineare Wellengleichungen. II,” Manuscripta Math.,20, No. 3, 227–244 (1977).
H. Pecher, “Ein nichlinearer Interpolationssatz und seine Anwendung auf nichlineare Wellengleichungen,” Math. Z.,161, No. 1, 9–40 (1978).
J. Rauch, “I. The u5 Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations,” in: Nonlinear Partial Differential Equations and Their Applications, Vol. I, Pitman, Boston (1981), pp. 335–364.
I. E. Segal, “The global Cauchy problem for a relativistic scalar field with power interaction,” Bull. Soc. Math. France,91, No. 2, 129–135 (1963).
W. A. Strauss, “On weak solutions of semi-linear hyperbolic equations,” An. Acad. Brasil. CiÊnc,42, No. 4, 645–651 (1970).
R. S. Strichartz, “Convolutions with kernels having singularities on a sphere,” Trans. Amer. Math. Soc,148, No. 2, 461–471 (1970).
R. S. Strichartz, “A priori estimates for the wave equation and some applications,” J. Funct. Anal.,5, No. 2, 218–235 (1970).
M. Struwe, “Globally regular solutions to the u5 Klein-Gordon equation,” Ann. Scuola Norm. Sup. Pisa,15, No. 3, 495–513 (1988).
H. Triebel, Theory of Function Spaces, BirkhÄuser, Basel (1983).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 24–64, 1990.
Rights and permissions
About this article
Cite this article
Kapitanskii, L.V. Cauchy problem for a semilinear wave equation. III. J Math Sci 62, 2619–2645 (1992). https://doi.org/10.1007/BF01102635
Issue Date:
DOI: https://doi.org/10.1007/BF01102635