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

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.

This is a continuation of the author’s paper Delayed Singularity Formation of Solutions of Nonlinear Wave Equations in Higher Dimensions, Comm. Pure Appl. 29, 1976, pp. 649–682, referred to as (*) in the sequel.