# Asymptotic Behavior for High Moments of the Fractional Heat Equation with Fractional Noise

• Litan Yan
• Xianye Yu
Article

## Abstract

In this paper, we investigate the large time behavior of the solution to the fractional heat equation
\begin{aligned} \frac{\partial u}{\partial t}(t,x)=-(-\Delta )^{\beta /2}u(t,x)+u(t,x)\frac{\partial ^{d+1} W}{\partial t\partial x_1\cdots \partial x_d},\quad t>0,\quad x\in \mathbb {R}^d, \end{aligned}
where $$\beta \in (0,2)$$ and the noise W(tx) is a fractional Brownian sheet with indexes $$H_0, H_1,\ldots ,H_d\in (\frac{1}{2},1)$$. By using large deviation techniques and variational method, we find a constant $$M_1$$ such that for any integer $$p\ge 1$$ and $$\alpha _0\beta +\alpha <\beta ,$$
\begin{aligned} \lim _{t\rightarrow \infty }t^{-\frac{2\beta -\beta \alpha _0-\alpha }{\beta -\alpha }} \log Eu(t,x)^p=p^{\frac{2\beta -\alpha }{\beta -\alpha }}\left( \frac{\alpha _H}{2} \right) ^{\frac{\beta }{\beta -\alpha }}M_1, \end{aligned}
where $$\alpha _0=2-2H_0$$, $$\alpha =\sum \nolimits _{j=1}^d(2-2H_j)$$ and $$\alpha _H=\prod \nolimits _{i=0}^dH_i(2H_i-1)$$.

## Keywords

Fractional heat equation Fractional Brownian sheet Asymptotic behavior

60H15 35B40

## Notes

### Acknowledgements

The first author was sponsored by National Natural Science Foundation of China (No. 11571071) and the Innovation Program of Shanghai Municipal Education Commission (12ZZ063). The second author was supported by National Natural Science Foundation of China (No. 11701589).

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