Large deviation principle for stochastic  heat equation  with general rough noise

Please see Manuscript PDF for complete Abstract with equations:We study Freidlin-Wentzell's large deviation principle for one dimensional nonlinear stochastic heat equation driven by a Gaussian noise: $$ \frac{\partial u^{\e}(t,x)}{\partial t}=\frac{\partial^2 u^{\e}(t,x)}{\partial x^2}+\sqrt{\e}\sigma(t, x, u^{\e}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in\RR, $$ where $\dot W$ is white in time and fractional in space with Hurst parameter $H\in(\frac 14,\frac 12)$. Recently, Hu and Wang ({\it Ann. Inst. Henri Poincar\'e Probab. Stat.} {\bf 58} (2022) 379-423) studied the well-posedness of this equation without the technical condition of $\sigma(0)=0$ which was previously assumed in Hu et al. ({\it Ann. Probab}. {\bf 45} (2017) 4561-4616). We adopt a new sufficient condition proposed by Matoussi et al. ({\it Appl. Math. Optim.} \textbf{83} (2021) 849-879) for the weak convergence criterion of the large deviation principle.