Almost global existence in the plane wave- nonlinear dielectric interaction problem

AbstractThe problem of the time evolution of the electromagnetic field in a plane wave propagating into a nonlinear dielectric, with nontrivial conduction vector governed by a nonlinear Ohm's law is considered. For the relevant, associated, inhomogeneous quasilinear hyperbolic system of balance laws, with associated periodic initial data, we establish the following result on almost global existence, in time, for solutions of the initial-value problem: either for sufficiently small initial data the maximal time of existence t∞ of a C1 solution is infinitely large or, for ∞ finite, no matter how small, but finite, the initial data is taken, for all initial data which are sufficiently small, t∞ is bounded below by a function of the initial data which increases without bound as the magnitude of the data goes to zero