  We discuss the nonlinear dynamics and fluctuations of interfaces with bending
rigidity under the competing attractions of two walls with arbitrary
permeabilities. This system mimics the dynamics of confined membranes. We use a
two-dimension hydrodynamic model, where membranes are effectively
one-dimensional objects. In a previous work [T. Le Goff et al, Phys. Rev. E 90,
032114 (2014)], we have shown that this model predicts frozen states caused by
bending rigidity-induced oscillatory interactions between kinks (or domain
walls). We here demonstrate that in the presence of tension, potential
asymmetry, or thermal noise, there is a finite threshold above which frozen
states disappear, and perpetual coarsening is restored. Depending on the
driving force, the transition to coarsening exhibits different scenarios.
First, for membranes under tension, small tensions can only lead to transient
coarsening or partial disordering, while above a finite threshold, membrane
oscillations disappear and perpetual coarsening is found. Second, potential
asymmetry is relevant in the non-conserved case only, i.e. for permeable walls,
where it induces a drift force on the kinks, leading to a fast coarsening
process via kink-antikink annihilation. However, below some threshold, the
drift force can be balanced by the oscillatory interactions between kinks, and
frozen adhesion patches can still be observed. Finally, at long times, noise
restores coarsening with standard exponents depending on the permeability of
the walls. However, the typical time for the appearance of coarsening exhibits
an Arrhenius form. As a consequence, a finite noise amplitude is needed in
order to observe coarsening in observable time.
