Fritz John pp 589-623

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

• Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

This paper deals with existence of solutions 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)
for large times 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)
We assume that the 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)
, so that (1a) goes over into the classical linear wave equation
$${a_{ik}}(0) = {\delta _{ik}}$$
(1c)
for “infinitesimal” u. The solution u of (1a) is to be found from initial conditions for t = 0.

## Bibliography

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