Nonlinear random perturbations of PDEs and quasi-linear equations in Hilbert spaces depending on a small parameter

TL;DR

This paper investigates the asymptotic behavior of solutions to quasi-linear parabolic PDEs in Hilbert spaces with small parameters, establishing a large deviations principle and providing an explicit action functional.

Contribution

It introduces a novel analysis of SPDEs derived from quasi-linear PDEs with small parameters, including a large deviations principle and explicit action functional description.

Findings

Large deviations principle established for the SPDE solutions

Explicit form of the action functional derived

Asymptotic behavior characterized as the small parameter approaches zero

Abstract

We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the corresponding SPDE and we study the asymptotic behavior of its solutions, depending on the small parameter. We show that a large deviations principle holds and we give an explicit description of the action functional.