We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $\alpha\in (0,2]$, $\xi>0$ is the noise level, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function satisfying some growth conditions and the noise term behaves in space like the Riez kernel and is possibly correlated in time and $D$ is the unit open ball centered at the origin in $\mathbb{R}^d$. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level $\xi$. On the other hand when the noise term behaves in time like the fractional Brownian motion with index $H\in (1/2,1)$, We also derive explicit bounds leading to a well-known intermittency property.

中文翻译：

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有界域中由时空有色噪声驱动的一类随机热方程的矩界

我们考虑分数随机热型方程 \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi \sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} 具有非负有界初始条件，其中$\alpha\在 (0,2]$, $\xi>0$ 是噪声水平, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ 是满足某些增长条件的全局 Lipschitz 函数，噪声项表现为在空间中，如 Riez 核，并且可能在时间上相关，$D$ 是以 $\mathbb{R}^d$ 中的原点为中心的单位开球。当噪声项在时间上不相关时，我们建立一个变化在这些方程的解的增长取决于噪声水平 $\xi$。另一方面，当噪声项在时间上表现得像指数为 $H\in (1/2,1)$ 的分数布朗运动时，我们还推导出了导致众所周知的间歇性属性的明确界限。