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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
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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”.

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

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