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Nonexplosion for a large class of superlinear stochastic parabolic equations, in arbitrary spatial dimension

Michael Salins, Yuyang Zhang

TL;DR

This work studies finite-time explosion for the stochastic heat equation $u_t = A u + \sigma(u)\dot{W}$ on bounded domains with self-adjoint elliptic operator $A$ and space-time colored noise, under Neumann, periodic, or Dirichlet boundaries. It identifies a critical growth exponent $\chi = 1 + \frac{1-\eta}{2\beta}$ and proves global (non-explosive) existence of the mild solution in arbitrary dimension, extending prior one-dimensional results to more general noise structures and boundary conditions, including the case when $\chi$ reaches the critical level. The approach blends $L^1$-norm martingale methods, sharp stochastic-convolution moment bounds via a Da Prato–Zabczyk factorization, and a Borel–Cantelli argument to control large excursions, yielding robust non-explosion results. These results generalize and unify earlier findings (e.g., Mueller and Franzova) to higher dimensions and broader classes of colored noise, with explicit examples illustrating the equality case for the growth exponent.

Abstract

This paper explores the finite time explosion of the stochastic parabolic equation $\frac{\partial u}{\partial t}(t,x)=Au(t,x)+σ(u(t,x))\dot{W}(t,x)$ in arbitrary bounded spatial domain with a large class of space-time colored noise under Neumann, periodic or Dirichlet boundary conditions where $A$ is second-order self-adjoint elliptic operator and $σ$ grows like $σ(u)\approx C(1+|u|^χ)$ where $χ=1+\frac{1-η}{2β}$ with $η$ and $β$ are the parameters related to the singularities of heat kernel and noise covariance kernel. We improve upon previous results by proving the theory in arbitrary spatial dimension, general elliptic operator, general space-time colored noise, a larger class of boundary conditions and proves that $χ$ can reach the level $1+\frac{1-η}{2β}$.

Nonexplosion for a large class of superlinear stochastic parabolic equations, in arbitrary spatial dimension

TL;DR

This work studies finite-time explosion for the stochastic heat equation on bounded domains with self-adjoint elliptic operator and space-time colored noise, under Neumann, periodic, or Dirichlet boundaries. It identifies a critical growth exponent and proves global (non-explosive) existence of the mild solution in arbitrary dimension, extending prior one-dimensional results to more general noise structures and boundary conditions, including the case when reaches the critical level. The approach blends -norm martingale methods, sharp stochastic-convolution moment bounds via a Da Prato–Zabczyk factorization, and a Borel–Cantelli argument to control large excursions, yielding robust non-explosion results. These results generalize and unify earlier findings (e.g., Mueller and Franzova) to higher dimensions and broader classes of colored noise, with explicit examples illustrating the equality case for the growth exponent.

Abstract

This paper explores the finite time explosion of the stochastic parabolic equation in arbitrary bounded spatial domain with a large class of space-time colored noise under Neumann, periodic or Dirichlet boundary conditions where is second-order self-adjoint elliptic operator and grows like where with and are the parameters related to the singularities of heat kernel and noise covariance kernel. We improve upon previous results by proving the theory in arbitrary spatial dimension, general elliptic operator, general space-time colored noise, a larger class of boundary conditions and proves that can reach the level .
Paper Structure (6 sections, 6 theorems, 86 equations)

This paper contains 6 sections, 6 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Results
  3. Examples
  4. The $L^1$ norm of $u(t,x)$
  5. Moment estimates on the stochastic convolution
  6. Non-explosion of $u(t,x)$

Key Result

Theorem 1

Let $p$ be large enough that $\frac{1+\beta}{p}<\frac{1-\eta}{2}-\frac{\beta}{p-2}$, where $\beta$ and $\eta$ are given in Assumption assumption3 in section 2 and assume that $\varphi(t,x)$ is adapted and bounded. Define the stochastic convolution There exists $C_p>0$, independent of $T>0$ and $\varphi$ such that

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Lemma 5