Blow-up estimates for a system of semilinear SPDEs driven by mixed fractional Brownian motions

TL;DR

This paper investigates the conditions under which solutions to a system of semilinear stochastic partial differential equations driven by mixed Brownian and fractional Brownian motions either blow up in finite time or exist globally, providing bounds and probabilistic estimates.

Contribution

It introduces new blow-up estimates and conditions for global solutions in systems driven by mixed stochastic processes, utilizing Malliavin calculus for probabilistic analysis.

Findings

Finite-time blow-up solutions are characterized with bounds.

Conditions for the existence of global weak solutions are established.

A lower bound for blow-up probability is derived using Malliavin calculus.

Abstract

In this paper, we obtain the existence and finite-time blow-up for the solution to a system of semilinear stochastic partial differential equations driven by a combination of Brownian and fractional Brownian motions. Under suitable assumptions, lower and upper bounds for the finite-time blow-up solution are obtained. We provide sufficient conditions for the existence of a global weak solution to the system. Further, a lower bound for the probability of the finite-time blow-up solution of the considered system is provided by using Malliavin calculus.