Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-09-02T04:43:50.107Z Has data issue: false hasContentIssue false
  • English
  • Français

A global existence theorem for the Cauchy problem of nonlinear wave equations

Published online by Cambridge University Press:  14 November 2011

Jianmin Gao  and
Lichen Xu
Jianmin Gao
Affiliation:
Department of Applied Mathematics, Tongji University, Shanghai, China
Lichen Xu
Affiliation:
Department of Applied Mathematics, Tongji University, Shanghai, China
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 3-4 , 1988 , pp. 261 - 269
Copyright
Copyright © Royal Society of Edinburgh 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Klainerman, S.. Long time behavior of solution to nonlinear evolution equations. Arch. Rational Mech. Anal. 78 (1982), 73–98.CrossRefGoogle Scholar
2Klainerman, S.. Uniform decay estimate and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math. 38 (1985), 321–332.CrossRefGoogle Scholar
3John, F.. Blow-up for quasi-linear wave equations in three space dimensions. Comtn. Pure Appl. Math. 34 (1981), 29–51.CrossRefGoogle Scholar
4Shatah, J.. Global existence of small solutions to nonlinear evolution equations. J. Differential Equations 46 (1982), 409–425.CrossRefGoogle Scholar
5Christodoulou, D.. Global solution of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39 (1986) 267–282.CrossRefGoogle Scholar
6Matsumura, A. and Nishida, T.. The initial value problem for the equation of motion of compressible viscous and heat conductive fluids. Proc. Japan Acad. Ser A Math. Sci. 55 (1979), 337–342.CrossRefGoogle Scholar
7Strauss, W. A.. Decay and asymptotics for . J. Funct. Anal. 2 (1968), 409–457.CrossRefGoogle Scholar
8Adams, R. A.. Sobolev spaces (New York: Academic Press, 1975).Google Scholar