Monotonicity of stable solutions in shadow systems

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.