Stabilization of oscillations in a phase transition model

Abstract

In this paper we analyze a model presenting formation of microstructure depending on the parameters and the initial data. In particular we investigate how the presence of stochastic perturbations affects this phenomenon in its asymptotic behavior. Two different sufficient conditions are provided in order to prevent the formation of microstructure: the first one for Stratonovich’s noise while the second for Itˆo’s noise. The main contribution of the paper is that these conditions are independent of the initial values unlike in the deterministic model. Thus, we can interpret our results as some kind of stabilization produced by both types of noise.