Do general perturbations in the Swift–Hohenberg equation settle to steady states?

Ascertain whether general perturbations of the unstable trivial equilibrium u ≡ 0 in the one-dimensional cubic Swift–Hohenberg equation ∂t u = −(1+∂x^2)^2 u + μ u − u^3 with μ > 0 generically converge to steady solutions (for example, stationary wavetrains) rather than exhibiting persistent non-stationary dynamics, and identify precise conditions under which convergence to steady states occurs.

Background

In the discussion of the onset of pattern formation, the authors explain that perturbations near the Turing instability lead to bounded growth and small-amplitude stationary periodic solutions (wavetrains). This is supported by the variational structure and dissipation of the Swift–Hohenberg equation, and by amplitude-equation analysis.

However, beyond sinusoidal or specially structured perturbations, it is not established whether arbitrary perturbations of the trivial state at μ > 0 will asymptotically settle into steady patterns. The authors explicitly note this uncertainty, highlighting a gap between observed behavior near onset and a general dynamical characterization of the long-time fate of generic perturbations.