[Paper Review] Global existence and non-existence of stochastic parabolic equations

Guangying Lv, Jinlong Wei|arXiv (Cornell University)|Feb 20, 2019

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes global existence and finite-time blowup results for stochastic parabolic equations on both bounded domains and the whole space using a novel stochastic concavity method that eliminates the need for non-negativity assumptions on solutions. It proves that for $1 < p \leq 1 + \frac{2}{d}$, nontrivial initial data lead to finite-time blowup of the expectation and mean square of solutions, even under stochastic noise, extending deterministic Fujita-type phenomena to the stochastic setting with new noise-induced effects.

ABSTRACT

This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing results, we delete the assumption that the solutions to stochastic heat equations are non-negative. Then the blowup phenomenon in the whole space is obtained by using the properties of heat kernel. We obtain that the solutions will blow up in finite time for nontrivial initial data.

Motivation & Objective

• To study the finite-time blowup phenomenon in stochastic parabolic equations on bounded domains and in the whole space.
• To remove the restrictive assumption that solutions must be non-negative, which is common in prior stochastic blowup analyses.
• To develop a new stochastic concavity method that avoids positivity requirements for proving blowup.
• To extend the deterministic Fujita phenomenon to stochastic parabolic equations with general noise structures.
• To establish conditions under which solutions blow up in finite time or exist globally, depending on nonlinearity, initial data, and noise impact.

Proposed method

• Introduce a stochastic concavity method to prove blowup without requiring solution positivity, replacing the traditional stochastic Kaplan method.
• Apply Itô’s formula to the $L^1$-norm of the negative part of the solution to show it vanishes almost surely, implying non-negativity is not required.
• Use comparison principles and heat kernel estimates to analyze blowup in the whole space, comparing stochastic solutions to deterministic counterparts.
• Establish mild solution representation using the heat kernel and stochastic convolution to analyze moments of solutions.
• Analyze the expectation and second moment of solutions via integral inequalities and comparison with deterministic equations.
• Use the properties of the heat kernel and stochastic integral bounds to prove blowup of $\mathbb{E}[u(x,t)]$ and $\mathbb{E}[|u(x,t)|^2]$ under suitable initial data and nonlinearity.

Experimental results

Research questions

• RQ1 Can the finite-time blowup of stochastic parabolic equations be proven without assuming non-negativity of solutions?
• RQ2 How does the presence of multiplicative noise affect the blowup time and existence of solutions in the whole space?
• RQ3 To what extent does the stochastic Fujita-type phenomenon differ from the deterministic case in terms of blowup conditions?
• RQ4 Can the stochastic concavity method be generalized to SPDEs with non-Lipschitz noise coefficients?
• RQ5 What is the role of initial data size and spatial decay in determining global existence versus finite-time blowup under stochastic perturbations?

Key findings

• A new stochastic concavity method is developed that proves finite-time blowup without requiring the solution to be non-negative, removing a key assumption in prior works.
• For $1 < p \leq 1 + \frac{2}{d}$, the expectation $\mathbb{E}[u(x,t)]$ blows up in finite time for any nontrivial initial data, even with noise.
• The second moment $\mathbb{E}[|u(x,t)|^2]$ also blows up in finite time when the initial data is sufficiently large and $m > 1$.
• The blowup result in the whole space is established via comparison with the deterministic heat equation and heat kernel estimates, extending Fujita’s results to the stochastic case.
• Noise can induce blowup even when the deterministic equation would have global solutions, depending on the nonlinearity and initial data size.
• The method applies to general multiplicative noise $\sigma(u)$, including non-Lipschitz cases, broadening the scope of existing blowup results.