References
[C] Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math.39, 267–282 (1986)
[CW] Chen, C., von Wahl, W.: Das Rand-Anfangswertproblem für quasilineare Wellengleichungen in Sobolevräumen niedriger Ordnung. J. Rein Angew. Math.337, 77–112 (1982)
[DH] Dafermos, C. M., Hrusa, W. J.: Energy methods for quasilinear hyperbolic initial-boundary value problems, applications to elastodynamics. Arch. Ration. Mech. Anal.87, 267–292 (1985)
[D] Dionne, P.-A.: Sur les problémes de Cauchy hyperboliques bien posés. J. Anal. Math.10, 1–90 (1962)
[EH] Ellis, G.F.R., Hawking, S.W.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973
[EM] Eardley, D.M., Moncrief, V.: The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space 1 and 2. Commun. Math. Phys.83, 171–212 (1982)
[F] Friedman, A.: Partial Differential Equations. New York Chicago San Francisco: Holt, Rinehart and Winston 1969
[GT] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, second ed. Berlin Heidelberg New York: Springer 1983
[HKM] Hughes, T.J.R., Kato, T., Marsden, J.E.: Well-posed quasilinear second order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal.63, 273–294 (1977).
[J] John, F.: Blow-up for quasilinear wave equations in three space dimensions. Commun. Pure Appl. Math.34, 29–51 (1981)
[K] Klainerman, S.: The null condition and global existence to nonlinear wave equations. Lect. Appl. Math.23, 293–326 (1986)
[Ka1] Kato, T.: Quasilinear equations of evolution, with applications to partial differential equations. In: Everitt, W.N. (ed.) Spectral theory and differential equations. Proc. 1974. (Lect. Notes Math., vol. 448, pp. 25–70) Berlin Heidelberg New York: Springer 1975
[Ka2] Kato, T.: Abstract differential equations and nonlinear mixed problems. (Lezione Fermiane). Pisa: Scuola Normale Superiore 1985
[Ko] Koch, H.: Hyperbolic equations of second order. Dissertation, Heidelberg (1990)
[KS] Kikuchi, M., Shibata, Y.: On the mixed problem for some quasilinear hyperbolic system with fully nonlinear boundary condition. J. Differ. Equations80, 154–197 (1989)
[L] Ladyzenskaja, O. A.: Linear and quasilinear equations of parabolic type. Providence, RI: Am. Math. Soc. 1985
[M1] Majda, A.: The stability of multi-dimensional shock fronts. Mem. Am. Math. Soc.275 (41) (1983)
[M2] Majda, A.: The existence of multi-dimensional shock fronts. Mem. Am. Math. Soc.281 (43) (1983)
[MT] Majda, A., Thomann, E.: Multi dimensional shock fronts for second order wave equations. Commun. Partial. Diffu. Equations12 (7), 777–828 (1987)
[NS] Nakamura, G., Shibata, Y.: On a local existence theorem of Neumann problem for some quasilinear hyperbolic systems of 2nd order. Math. Z.202, 1–64 (1989)
[S1] Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Commun. Pure Appl. Math.38, 685–696 (1985)
[S2] Shatah, J.: Weak solutions and development of singularities of the SU(2) σ-model. Commun. Pure Appl. Math.41, 459–479 (1988)
[Sh1] Shibata, Y.: On the Neumann problem for some linear hyperbolic systems of 2nd order. Tsukuba J. Math.12(1), 149–209 (1988)
[Sh1] Shibata, Y.: On the Neumann problem for some linear hyperbolic systems of 2nd order with coefficients in Sobolev spaces. Tsukuba J. Math.13, (2), 283–352 (1989)
[SS] Simpson, H.C., Spectro, S.J.: On the positivity of the second variation in finite elasticity. Arch. Ration. Mech. Anal.98, 1–30 (1987)
[S] Struwe, M.: Globally regular solutions to the u5 Klein-Gordon equation. Ann. Sc. Norm. Superl. Pisa, Sc. Fis. Mat., IV. Ser.15, Fasc 3 (1988)
[S] Shibata, Y., Tsutsumi, Y.: Local existence of solution for the initial boundary value problem of fully nonlinear wave equation. Nonl. Anal.11 (3), 335–365 (1987)
[W] Weidemaier, P.: Existence of regular solutions for a quasilinear wave equation with the third boundary condition. Math. Z.191, 449–465 (1986)
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Koch, H. Mixed problems for fully nonlinear hyperbolic equations. Math Z 214, 9–42 (1993). https://doi.org/10.1007/BF02572388
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DOI: https://doi.org/10.1007/BF02572388