Extending our previous work in the strictly parabolic case, we show that a
linearly unstable Lax-type viscous shock solution of a general quasilinear
hyperbolic--parabolic system of conservation laws possesses a
translation-invariant center stable manifold within which it is nonlinearly
orbitally stable with respect to small L1∩H3 perturbations, converging
time-asymptotically to a translate of the unperturbed wave. That is, for a
shock with p unstable eigenvalues, we establish conditional stability on a
codimension-p manifold of initial data, with sharp rates of decay in all
Lp. For p=0, we recover the result of unconditional stability obtained by
Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case
is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p