Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on \(\mathbb {R}\) Driven by Space–Time White Noise

In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line \(\mathbb {R}\) $\mathbb{R}$ driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space \(C([0,T], C_\textrm{tem}(\mathbb {R}))\) $C([0,T],C_{\text{tem}}(\mathbb{R}))$ , where \(C_\textrm{tem}(\mathbb {R}):=\{f\in C(\mathbb {R}): \sup _{x\in \mathbb {R}} \left( |f(x)|e^{-\lambda |x|}\right) <\infty \text { for any } \lambda >0\}\) $C_{\text{tem}}(\mathbb{R}):=\{f\in C(\mathbb{R}):{sup}_{x\in \mathbb{R}}\left(|f(x)|,e^{-\lambda |x|}\right)<\infty \text{for any}\lambda >0\}$ . Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space \(C([0,T], C_\textrm{tem}(\mathbb {R}))\) $C([0,T],C_{\text{tem}}(\mathbb{R}))$ , one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role.