A shadow system appears as a limit of a reaction-diffusion system in which
some components have infinite diffusivity. We investigate the spatial struc
ture of its stable solutions. It is known that, unlike scalar reaction-diff
usion equations, some shadow systems may have stable nonconstant (monotone)
solutions. On the other hand, it is also known that in autonomous shadow s
ystems any nonconstant non-monotone stationary solution is necessarily unst
able. In this paper, it is shown in a general setting that any stable bound
ed (not necessarily stationary) solution is asymptotically homogeneous or e
ventually monotone in x.