Long time existence of solutions for \({\partial{} ^ 2 \over{} \partial{} T^ 2} u(x,t) + {\partial{} \over{} \partial{} T} \alpha{} (u(x,t)) = {\partial{} ^ 2 \over{} \partial{} X^ 2} \beta{} (u(x,t))\) (Q1174527)

The author investigates the long time existence of a classical solution of a nonlinear wave equation of the form \[ u_{tt}(x,t)+\alpha(u(x,t))_ t=\beta(u(x,t))_{xx}, x\in R^ 1, t\geq0 \] arising in various problems of nonlinear electromagnetics. Under fairly weak assumptions on the functions \(\alpha\) and \(\beta\) it is shown that for all sufficiently small initial data the maximal time of existence of a \(C^ 1\) solution is bounded from below by a function of the initial data which increases without bound as the magnitude of the initial data goes to zero. The proof is based on Riemann invariants and approximate energy estimates.