  Motivated by the study of conditional stability of traveling waves, we give
an elementary $H^2$ center stable manifold construction for quasilinear
parabolic PDE, sidestepping apparently delicate regularity issues by the
combination of a carefully chosen implicit fixed-point scheme and
"damping-type" $H^s$ energy estimates of a type familiar from the study of
hyperbolic--parabolic and relaxation systems. An important feature of these
methods is that they generalize to situations such as the hyperbolic--parabolic
or relaxation case for which parabolic-type smoothing estimates are
unavailable. As an application, we show conditional stability of Lax- or
undercompressive shock waves of general quasilinear parabolic systems of
conservation laws by a pointwise stability analysis on the center stable
manifold.
