Fritz John pp 589-623 | Cite as

# Lower Bounds for the Life Span of Solutions of Nonlinear Wave Equations in Three Dimensions

Chapter

## Abstract

This paper deals with existence of solutions for large times We assume that the , so that (1a) goes over into the classical linear wave equation for “infinitesimal”

*u*(*x*_{1},*x*_{2},*x*_{3}) =*u*(*x, t*) of a nonlinear wave equation of the form$${u_{tt}} - \sum\limits_{i,k = 1}^3 {{a_{ik}}} (u\prime ){u_{{x_i}}}_{{x_k}} = 0$$

(1a)

*t*. Here*u*′ stands for the gradient vector$$u\prime = ({u_{{x_1}}},{u_{{x_2}}},{u_{{x_3}}},{u_t}) = ({D_1}u,{D_2}u,{D_3}u,{D_4}u) = Du$$

(1b)

*a*_{ ik }(*U*) are in*C*^{∞}in a closed ball |*U*| in ≦*δ*in ℝ^{4}, and that$$u = {u_{tt}} - \Delta u = 0$$

(1d)

$${a_{ik}}(0) = {\delta _{ik}}$$

(1c)

*u*. The solution*u*of (1a) is to be found from initial conditions for*t*= 0.## Preview

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## Bibliography

- [1]Klainerman, S.,
*Global existence for nonlinear wave equations*, Comm. Pure Appl. Math. 33, 1980, pp. 43–101.CrossRefGoogle Scholar - [2]Shatah, J.,
*Global existence of small solutions to nonlinear evolution equations*, preprint.Google Scholar - [3]Klainerman, S., and Ponce, G.,
*Global, small amplitude solutions to nonlinear evolution equations*, preprint.Google Scholar - [4]John, F.,
*Blow-up for quasi-linear wave equations in three space dimensions*, Comm. Pure Appl. Math. 34, 1981, pp. 29–51.CrossRefGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1983