We investigate an initial-boundary value problem for a quasilinear
nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary
condition of Silver & M\"uller type in a smooth, bounded, strictly star-shaped
domain of R3. Imposing usual smallness assumptions in addition to
standard regularity and compatibility conditions, a nonlinear stabilizability
inequality is obtained by showing nonlinear dissipativity and
observability-like estimates enhanced by an intricate regularity analysis. With
the stabilizability inequality at hand, the classic nonlinear barrier method is
employed to prove that small initial data admit unique classical solutions that
exist globally and decay to zero at an exponential rate. Our approach is based
on a recently established local well-posedness theory in a class of
H3-valued functions.Comment: 22 page