ABSTRACT
The natural counterpart to the problem of shock formation in the solutions of wave-dielectric interaction problems in one-space dimension. the proof of existence of weak solutions, for both the wave-dielectric system and the nonlinear transmission line system, requires, first, a preliminary discussion of basic concepts related to the notions of weak convergence, compensated compactness, and the Young measure; in connection with our discussion of the distributed parameter nonlinear transmission line problem. The chapter describes how the basic type of energy argument can be applied to study the global existence and asymptotic stability of classical solutions to the initial-value problem associated with the wave-dielectric interaction system when the initial data is small in some appropriate sense. It begins with the discussion of global existence of smooth solutions for the one-space dimension wave-dielectric interaction problem under the assumption of small initial data.
