Finite time blowup of the stochastic shadow Gierer-Meinhardt System

Fang Li,Lihu Xu

doi:10.1214/ecp.v20-4298

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.

Highlights

Inspired by the recent work [9] and [10], we study the blow up of the shadow Gierer-Meinhardt system with random migrations with the following form: ∂tu = ∆u − u + up ξq dξ = −ξ + ur ξs dt + ξdBt ∂u ∂ν = u(0) = u0ξ(0) = ξ0, in O × (0, T ), in (0, T ), on ∂O × (0, T ), in O, (1.1)
By choosing some special initial data, we prove that with probability 1, stochastic shadow Gierer-Meinhardt system blows up in finite time in the pointwise sense
We give a upper bound for the blowup time and some estimates about this bound

Summary

Introduction

By choosing some special (random) initial data, we prove that with probability 1, stochastic shadow Gierer-Meinhardt system blows up in finite time in the pointwise sense. We give a (random) upper bound for the blowup time and some estimates about this bound. Blowup of stochastic Gierer-Meinhardt system blows up in the pointwise sense if we choose some suitable (random) initial data. Itô’s formula in the proof of Lemma 2.2 below is the key point for finding a monotone stochastic process ξ(t), which paves the way to applying classical PDE techniques and estimating the upper bounds of blow up time.

Results

Conclusion