Global Nonexistence of Smooth Electric Induction Fields in Nonlinear Dielectrics. I. Infinite Cylindrical Dielectrics

摘要

Coupled nonlinear wave equations are derived for the evolution of the components of the electric induction field equivalent D in a class of rigid nonlinear dielectrics governed by the nonlinear constitutive relation equivalent E = Lambda (d) equivalent D, where equivalent E is the electric field and Lambda > 0 is a scalar-valued vector function. For the special case of an infinite one-dimensional dielectric rod, embedded in a perfect conductor, it is shown that, under relatively mild conditions on Lambda, solutions of the corresponding initial-boundary value problem for the electric induction field cannot exist globally in time in L2 sense if it is assumed that the electric field in the rod is perpendicular to the axis of the rod and varies as the coordinate along that axis. (Author)