噪声对随机热方程不爆破的作用

中文摘要 AI 辅助

考虑热方程：\begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\\,dt+b(u(t,x))\\,dt, &t>0,\\ x\in D,\\\\[1mm] u(t,x)=0,&t\ge0,\\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} 众所周知，系数$b$的超线性增长会导致解$u$在有限时间内爆破。在本文中，我们证明随机噪声将阻止解的爆炸。更精确地，我们证明随机热方程：\begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\\,dt+b(u(t,x))\\,dt+u(t,x)^n\\,dB_t, &t>0,\\ x\in D,\\\\[1mm] u(t,x)=0,&t\ge0,\\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} 存在全局解，如果 \begin{equation} z\\,b(z)\leq C_b(1+z^2)+\eta |z|^{2n}, \qquad z\in\R. \end{equation} 其中$\eta\in [0, \frac{1}{2})$，这里$n$是任意但固定的整数，$B_t, t\geq 0$是布朗运动。截断和比较技术起着重要作用。

英文摘要

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12Δu(t,x)\,dt+b(u(t,x))\,dt, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\R\to\R$, there always exists a locally Lipschitz noise coefficient $σ:\R\to\R$ such that the stochastic heat equation \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12Δu(t,x)\,dt+b(u(t,x))\,dt+σ(u(t,x))\,dB_t, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a unique global solution for every $u_0\in C_c(D)$. Here $(B_t)_{t\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $σ$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.

发表机构

• Hefei University of Technology（合肥工业大学）
• University of Science and Technology of China（中国科学技术大学）
• University of Manchester（曼彻斯特大学）

机构由 AI 辅助整理，请以论文原文为准。

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13 pages