An initial value problem for Maxwell’s equations is considered in a medium of constant dielectricity and conductivity filling \({\mathbb{R}}^ 3\). The magnetic field H is assumed to be a nonlinear, monotone function \(\zeta\) : \({\mathbb{R}}^ 3\to {\mathbb{R}}^ 3\) of the magnetic induction B. Moreover, \(\zeta\) is assumed to be the derivative of a convex function. In this case the system can easily be seen to be symmetric hyperbolic and Kato’s results guarantee under suitable conditions on \(\zeta\), assuming smooth divergence free initial data and making use of regularity results for solutions of Maxwell’s equations that a unique smooth solution exists at least locally.
The aim of the paper is to show that for small initial values the local solutions can be continued to yield a global existence result. This is achieved by noting that introducing vector potentials Maxwell’s equations can under suitable assumptions be written as a second order system and by extending results of A. Matsumura [Publ. Res. Inst. Math. Sci., Kyoto Univ. 13, 349-379 (1977; Zbl 0371.35030)] to the system of Maxwell’s equations.