Quenching time and probability estimates for a stochastic reaction-diffusion system with coupled inner singular absorption terms driven by mixed noises

TL;DR

This paper analyzes a stochastic reaction-diffusion system with mixed noise, deriving bounds on quenching time and probability, and validating findings through numerical simulations, advancing understanding of noise effects on quenching phenomena.

Contribution

It provides explicit bounds on quenching time and probability for a stochastic PDE with mixed noise, using Malliavin calculus and numerical validation, which is novel in the context of coupled singular absorption terms.

Findings

Explicit bounds for quenching time derived

Quantifiable bounds on quenching probability established

Numerical results validate theoretical predictions

Abstract

This paper investigates a stochastic parabolic system under Robin boundary conditions, for which the deterministic counterpart exhibits finite quenching. The stochastic system incorporates mixed noise, combining standard one-dimensional Brownian motion and fractional Brownian motion. Under appropriate assumptions, we derive explicit lower and upper bounds for the quenching time of the solution and establish the global existence of a weak solution. Leveraging Malliavin calculus, we further obtain a quantifiable lower and upper bound on the quenching probability. To complement the theoretical analysis, we design a numerical scheme tailored to the system and present results that validate the analytical predictions, offering insights into the interplay between noise and quenching behaviour.