Osaka Journal of Mathematics

Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds

Soichiro Katayama and Kazuyoshi Yokoyama

Full-text: Open access

Abstract

We give a global existence theorem to systems of quasilinear wave equations in three space dimensions, especially for the multiple-speed cases. It covers a wide class of quadratic nonlinearities which may depend on unknowns as well as their first and second derivatives. Our proof is achieved through total use of pointwise and $L^2$-estimates concerning unknowns and their first and second derivatives.

Article information

Source
Osaka J. Math., Volume 43, Number 2 (2006), 283-326.

Dates
First available in Project Euclid: 6 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1152203942

Mathematical Reviews number (MathSciNet)
MR2262337

Zentralblatt MATH identifier
1195.35225

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35L05: Wave equation 35L15: Initial value problems for second-order hyperbolic equations 35L55: Higher-order hyperbolic systems

Citation

Katayama, Soichiro; Yokoyama, Kazuyoshi. Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds. Osaka J. Math. 43 (2006), no. 2, 283--326. https://projecteuclid.org/euclid.ojm/1152203942


Export citation

References

  • R. Agemi and K. Yokoyama: The null condition and global existence of solutions to systems of wave equations with different speeds; in Advances in Nonlinear Partial Differential Equations and Stochastics, S. Kawashima and T. Yanagisawa (Eds.), Series on Adv. Math. for Appl. Sci. 48, World Scientific, 1998, 43–86.
  • F. Asakura: Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimensions, Comm. in Partial Differential Equations 11 (1986), 1459–1487.
  • D. Christodoulou: Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), 267–282.
  • K. Hidano: The global existence theorem for quasi-linear wave equations with multiple speeds, Hokkaido Math. J. 33 (2004), 607–636.
  • L. Hörmander: Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Berlin, 1997.
  • S. Katayama: Global existence for a class of systems of nonlinear wave equations in three space dimensions, Chinese Ann. Math. 25B (2004), 463–482.
  • S. Katayama: Global and almost-global existence for systems of nonlinear wave equations with different propagation speeds, Diff. Integral Eqs. 17 (2004), 1043–1078.
  • S. Katayama: Global existence for systems of wave equations with nonresonant nonlinearities and null forms, J. Differential Equations 209 (2005), 140–171.
  • S. Klainerman: The null condition and global existence to nonlinear wave equations, Lectures in Appl. Math. 23 (1986), 293–326.
  • S. Klainerman and T.C. Sideris: On almost global existence for nonrelativistic wave equations in $3\mathrm{D}$, Comm. Pure Appl. Math. 49 (1996), 307–321.
  • M. Kovalyov: Resonance-type behaviour in a system of nonlinear wave equations, J. Differential Equations 77 (1989), 73–83.
  • K. Kubota and K. Yokoyama: Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation, Japanese J. Math. 27 (2001), 113–202.
  • H. Lindblad: On the lifespan of solutions of nonlinear wave equations with small initial data, Comm. Pure Appl. Math. 43 (1990), 445–472.
  • M. Ohta: Counterexample to global existence for system of nonlinear wave equations with different propagation speeds, Funkcialaj Ekvacioj, 46 (2003), 471–477.
  • T.C. Sideris and Shun-Yi Tu: Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33 (2001), 477–488.
  • C.D. Sogge: Global existence for nonlinear wave equations with multiple speeds; in Harmonic Analysis at Mount Holyoke, W. Beckner et al. (Eds.), Contemp. Math. 320, Amer. Math. Soc., Providence, RI, 2003, 353–366.
  • K. Yokoyama: Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan 52 (2000), 609–632.