Impact of noise on parabolic equations

TL;DR

This paper investigates the blowup behavior of stochastic parabolic equations, analyzing the probability of solutions remaining positive, blowing up, and the timing of blowup compared to deterministic cases.

Contribution

It provides new probabilistic insights into blowup phenomena and the timing of blowup events in stochastic parabolic equations, extending understanding beyond deterministic models.

Findings

Probability of solutions remaining positive is characterized.

Probability of finite-time blowup is quantified.

Comparison of blowup times with deterministic cases is established.

Abstract

In this short paper, we focus on the blowup phenomenon of stochastic parabolic equations. We first discuss the probability of the event that the solutions keep positive. Then, the blowup phenomenon in the whole space is considered. The probability of the event that the solutions blow up in finite time is given. Lastly, we obtain the probability of the event that blowup time of stochastic parabolic equations large than or less than the deterministic case.