Conditional stability of unstable viscous shocks

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Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L¹∩ H² perturbatoins, 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 Howard, Mascia, and Zumbrun.