Large time behavior of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions

TL;DR

This paper investigates the finite-time blowup behavior of semilinear stochastic PDEs driven by combined Brownian and fractional Brownian motions, analyzing how parameters influence blowup probability and timing.

Contribution

It provides new insights into the blowup phenomena of stochastic PDEs with mixed noise, including probability estimates and the role of eigenfunctions and parameters.

Findings

Derived estimates for blowup probability before a fixed time.

Identified influence of eigenvalues and parameters on blowup occurrence.

Analyzed the impact of mixed stochastic perturbations on PDE solutions.

Abstract

We study the trajectorywise blowup behavior of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are the mild and weak form of an associated random partial differential equation.