Semimartingale attractors for Allen-Cahn SPDEs driven by space-time white noise I: Existence and finite dimensional asymptotic behavior

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to Allen-Cahn type SPDEs driven by space-time white noise, establishing the existence of semimartingale attractors with finite-dimensional properties.

Contribution

It introduces the concept of semimartingale attractors for second order Allen-Cahn SPDEs and proves their existence, uniqueness, regularity, and finite-dimensional asymptotic behavior.

Findings

Existence of semimartingale global attractors for Allen-Cahn SPDEs.

Finite fractal dimension of the attractors.

Results on determining modes in both forward and pullback senses.

Abstract

We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa \cite{AL0}. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE 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 this random attractor and give a result on determining modes, both in the forward and the pullback sense.