SEMIMARTINGALE ATTRACTORS FOR ALLEN–CAHN SPDEs DRIVEN BY SPACE–TIME WHITE NOISE I: EXISTENCE AND FINITE DIMENSIONAL ASYMPTOTIC BEHAVIOR

Abstract

We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa [5]. In this paper we focus on second-order SPDEs of the Allen–Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and for the corresponding random PDEs of Allen–Cahn type; we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite-dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of these random attractors and give a result on determining modes, both in the forward and the pullback senses.