If a given solution of a PDE is stable, then, roughly
speaking, any other solution that starts near it, stays near it for
all time. This is an important concept in applications, because it is
typically only the stable solutions that are observed in practice. I
will outline two key mathematical difficulties that one can encounter
when analyzing the stability of time-periodic solutions of dissipative
PDEs on unbounded domains. Briefly, they are the presence of zero
eigenvalues that are embedded in the continuous spectrum and the time-
periodicity of the associated linear operator. In the context of
viscous shocks in systems of conservation laws, I will show how these
difficulties can be overcome. The method involves the development of a
contour integral representation of the linear evolution, similar to
that of a strongly continuous semigroup, and detailed pointwise
estimates on the resultant Greens function, which are sufficient for
proving nonlinear stability under the necessary assumption of spectral
stability.