Large-time behavior of weak solution to a Cauchy problem for the Swift-Hohenberg equation

We shall study the large-time behavior of weak solution to the Cauchy problem of the Swift-Hohenberg equation in both cases, the non-conserved and the conserved one, respectively. We first prove the existence of global weak solution to this Cauchy problem by employing the method of continuation of local solution. Then we apply spectral analysis to investigate time-decay rates of the solutions which are optimal in the sense that the rates coincide with those of the corresponding linear problems. The key ingredient in the proof is the delicate analysis of the ranges of frequencies.