In this paper, we investigate the large time behavior of the solution to the fractional heat equation ∂u∂t(t,x)=-(-Δ)β/2u(t,x)+u(t,x)∂d+1W∂t∂x1⋯∂xd,t>0,x∈Rd,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{\partial u}{\partial t}(t,x)=-(-\Delta )^{\beta /2}u(t,x)+u(t,x)\frac{\partial ^{d+1} W}{\partial t\partial x_1\cdots \partial x_d},\quad t>0,\quad x\in \mathbb {R}^d, \end{aligned}$$\end{document}where β∈(0,2)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in (0,2)$$\end{document} and the noise W(t, x) is a fractional Brownian sheet with indexes H0,H1,…,Hd∈(12,1)\documentclass[12pt]{minimal}
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\begin{document}$$H_0, H_1,\ldots ,H_d\in (\frac{1}{2},1)$$\end{document}. By using large deviation techniques and variational method, we find a constant M1\documentclass[12pt]{minimal}
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\begin{document}$$M_1$$\end{document} such that for any integer p≥1\documentclass[12pt]{minimal}
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\begin{document}$$p\ge 1$$\end{document} and α0β+α<β,\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _0\beta +\alpha <\beta ,$$\end{document}limt→∞t-2β-βα0-αβ-αlogEu(t,x)p=p2β-αβ-ααH2ββ-αM1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{t\rightarrow \infty }t^{-\frac{2\beta -\beta \alpha _0-\alpha }{\beta -\alpha }} \log Eu(t,x)^p=p^{\frac{2\beta -\alpha }{\beta -\alpha }}\left( \frac{\alpha _H}{2} \right) ^{\frac{\beta }{\beta -\alpha }}M_1, \end{aligned}$$\end{document}where α0=2-2H0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _0=2-2H_0$$\end{document}, α=∑j=1d(2-2Hj)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =\sum \nolimits _{j=1}^d(2-2H_j)$$\end{document} and αH=∏i=0dHi(2Hi-1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _H=\prod \nolimits _{i=0}^dH_i(2H_i-1)$$\end{document}.
