Asymptotic Dynamics of Stochastic $p$-Laplace Equations on Unbounded Domains

TL;DR

This thesis investigates the long-term behavior of solutions to stochastic p-Laplace equations with non-autonomous forcing on unbounded domains, establishing existence, uniqueness, and stability of random attractors under different noise conditions.

Contribution

It introduces new tail estimates to handle non-compactness and proves the existence and upper semicontinuity of random attractors for stochastic p-Laplace equations on unbounded domains.

Findings

Existence and uniqueness of random attractors for both additive and multiplicative noise cases.

Proved upper semicontinuity of attractors as noise intensity tends to zero.

Established asymptotic compactness of solution operators in unbounded domains.

Abstract

This thesis is concerned with the asymptotic behavior of solutions of stochastic $p$-Laplace equations driven by non-autonomous forcing on $\mathbb{R}^n$. Two cases are studied, with additive and multiplicative noise respectively. Estimates on the tails of solutions are used to overcome the non-compactness of Sobolev embeddings on unbounded domains, and prove asymptotic compactness of solution operators in $L^2(\mathbb{R}^n)$. Using this result we prove the existence and uniqueness of random attractors in each case. Additionally, we show the upper semicontinuity of the attractor for the multiplicative noise case as the intensity of the noise approaches zero.