This chapter deals with electromagnetic fields in complex media in the time domain. First, the chapter discusses the Maxwell equations in complex media in the time domain and shows that they can be expressed as integrodifferential equations. It then provides a convenient functional setting that allows the Maxwell equation to be treated as an integrodifferential evolution equation, and also provides some solvability and well posedness results for this problem based on a semigroup approach. Two alternative approaches to the solvability of evolution problems in the time domain, namely, the evolution family approach and an approach using finite-dimensional approximations (Faedo-Galerkin) approach, are then discussed. Finally, the chapter presents some extensions related to evolution problems.
Keywords: time domain, well posedness, integrodifferential equations, Maxwell equations, integrodifferential evolution equation, solvability, semigroup approach, evolution problems, evolution family approach, Faedo-Galerkin approach
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