In this work we deal with quasilinear Maxwell system
\[
\begin{cases}
\overset{\partial t\left(\epsilon_{0}E+\Phi\left(E\right)\right)=curl\: H,}{\partial_{t}H=-curlE,}\end{cases}
\]
where $\epsilon_{0}$=diag $\left(a^{2},b^{2},b^{2}\right)$ is a
diagonal matrix and $\Phi$ is a smooth matrix such that $\mid\Phi\mid$
has polynomial growth near E = O. Under suitable hypotheses on $\Phi$
we establish a global existence result for small amplitude solutions.
The main argument is the study of pseudo-differential equations obtained
diagonalizing the system and using for these equations a particular
von Wahl-type estimate described in our previous paper $\left[5\right]$.
