Finite time blowup of the stochastic shadow Gierer-Meinhardt System

TL;DR

This paper proves that the stochastic shadow Gierer-Meinhardt system can blow up in finite time with probability one, providing bounds on blowup time and showing how initial data amplitude influences blowup timing.

Contribution

It introduces a probabilistic analysis of finite-time blowup in the stochastic shadow Gierer-Meinhardt system, including bounds and effects of initial data amplitude.

Findings

System blows up in finite time with probability 1

Provides an upper bound for blowup time

Higher initial amplitude leads to faster blowup with positive probability

Abstract

By choosing some special (random) initial data, we prove that with probability $1$, the stochastic shadow Gierer-Meinhardt system blows up pointwisely in finite time. We also give a (random) upper bound for the blowup time and some estimates about this bound. By increasing the amplitude of the initial data, we can get the blowup in any short time with positive probability.