标题： 空间维数为$(d+1)$的时空分数阶随机偏微分方程的间歇性前缘 显示英文标题

标题： Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions

摘要： 我们考虑时间分数阶随机热方程$$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$在$(d+1)$维空间中，其中$\nu>0$，$\beta\in (0,1)$，$\alpha\in (0,2]$，$d<\min\{2,\beta^{-1}\}\a$，$\partial^\beta_t$是 Caputo 分数阶导数，$-(-\Delta)^{\alpha/2} $是各向同性稳定过程的生成元，$\stackrel{\cdot}{W}(t,x)$是时空白噪声，$\sigma:\R \to\RR{R}$是 Lipschitz 连续的。 米杰纳和纳内 在\cite{JebesaAndNane1}中证明了：(i) 该方程解的绝对矩呈指数增长；(ii) 这些矩最远的高尖峰到原点的距离随时间精确线性增长。 最后一个结果是在假设$\alpha=2$和$d=1.$的条件下证明的。 在本文中，我们将这一结果扩展到情况$\alpha=2$和$d\in\{1,2,3\}.$。 显示英文摘要

摘要： We consider time fractional stochastic heat type equation $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$, $d<\min\{2,\beta^{-1}\}\a$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $\sigma:\R \to\RR{R}$ is Lipschitz continuous. Mijena and Nane proved in \cite{JebesaAndNane1} that : (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions $\alpha=2$ and $d=1.$ In this paper we extend this result to the case $\alpha=2$ and $d\in\{1,2,3\}.$