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Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 9
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Original Articles

Local well-posedness of a quasilinear wave equation

Willy DörflerDepartment of Mathematics, Karlsruhe Institute of Technology, 76128Karlsruhe, Germany.View further author information
,
Hannes GernerDepartment of Mathematics, Karlsruhe Institute of Technology, 76128Karlsruhe, Germany.View further author information
&
Roland SchnaubeltDepartment of Mathematics, Karlsruhe Institute of Technology, 76128Karlsruhe, Germany.Correspondence [email protected]
View further author information
Pages 2110-2123 | Received 25 Jul 2014, Accepted 28 Aug 2015, Published online: 23 Sep 2015

References

  • Moloney J, Newell A. Nonlinear optics. Boulder (CO): Westview Press; 2004.
  • Pototschnig M, Niegemann J, Tkeshelashvili L, Busch K. Time-domain simulations of the nonlinear Maxwell equations using operator-exponential methods. IEEE Trans. Antennas Propag. 2009;57:475–483.
  • Kato T. Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral theory and differential equations (Proc. Dundee, 1974). Vol. 448, Lecture Notes in Mathamatics. Berlin: Springer; 1975. pp. 25–70.
  • Hughes TJR, Kato T, Marsden JE. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal. 1977;63:273–294.
  • Kato T. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 1975;58:181–205.
  • Sogge CD. Lectures on non-linear wave equations. 2nd ed. Boston (MA): International Press; 2008.
  • Smith HF, Tataru D. Sharp local well-posedness results for the nonlinear wave equation. Ann. Math. (2). 2005;2:291–366.
  • Metcalfe J, Sogge CD. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete Contin. Dyn. Syst. 2010;28:1589–1601.
  • Nakao M. On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains. J. Math. Soc. Jpn. 2003;55:765–795.
  • Weidemaier P. Existence of regular solutions for a quasilinear wave equation with the third boundary condition. Math. Z. 1986;191:449–465.
  • Yao P-F. Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Differ. Equ. 2007;241:62–93.
  • Kato T. Linear evolution equations of hyperbolic type. J. Fac. Sci. Univ. Tokyo, Sect. I A. 1970;17:241–258.
  • Amann H. Linear and quasilinear parabolic problems. Vol. I, Monographs in mathematics. Vol. 89, Boston (MA): Birkhäuser Boston Inc.; 1995.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer; 1983.
  • Fattorini HO. The Cauchy problem. Cambridge: Cambridge University Press; 1983.

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