  Under natural spectral stability assumptions motivated by previous
investigations of the associated spectral stability problem, we determine sharp
$L^p$ estimates on the linearized solution operator about a multidimensional
planar periodic wave of a system of conservation laws with viscosity, yielding
linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d
\ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and
$L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in
general be rather complicated, involving both convective (i.e., wave-like) and
diffusive effects.
