Spectral analysis for transition front solutions in multidimensional Cahn-Hilliard systems

We consider the spectrum associated with the linear operator obtained when a Cahn-Hilliard system on Rn is linearized about a planar transition front solution. In the case of single Cahn-Hilliard equations on Rn, it's known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate |ξ|3, where ξ is the Fourier transform variable corresponding with components transverse to the wave. Moreover, it has recently been verified that for single equations this spectral behavior implies nonlinear stability. In the current analysis, we establish that the same cubic rate law holds for a broad range of multidimensional Cahn-Hilliard systems. The analysis of nonlinear stability will be carried out separately.