  A subcritical pattern-forming system with nonlinear advection in a bounded
domain is recast as a slow-fast system in space and studied using a combination
of geometric singular perturbation theory and numerical continuation. Two types
of solutions describing the possible location of stationary fronts are
identified, whose origin is traced to the onset of convective and absolute
instability when the system is unbounded. The former are present only for
nonzero upstream boundary conditions and provide a quantitative understanding
of noise-sustained structures in systems of this type. The latter correspond to
the onset of a global mode and are present even with zero upstream boundary
condition. The role of canard trajectories in the nonlinear transition between
these states is clarified and the stability properties of the resulting spatial
structures are determined. Front location in the convective regime is highly
sensitive to the upstream boundary condition and its dependence on this
boundary condition is studied using a combination of numerical continuation and
Monte Carlo simulations of the partial differential equation. Statistical
properties of the system subjected to random or stochastic boundary conditions
at the inlet are interpreted using the deterministic slow-fast
spatial-dynamical system.
