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Consider the stochastic reaction-diffusion equation with logarithmicnonlinearity driven by space-time white noise: \begin{align}\label{1.a} \left\{\begin{aligned} & \mathrm{d}u(t,x) = \frac{1}{2}\Delta u(t,x)\,\mathrm{d}t+b(u(t,x)) \,\mathrm{d}t \nonumber\\ &  + \sigma(u(t,x))\,W(\mathrm{d}t,\mathrm{d}x), \ t>0, x\in I , \\ & u(0,x)=u_0(x), \quad x\in I.\nonumber \end{aligned} \right. \end{align} When $I$ is a compact interval,say $I=[0,1]$, the well-posedness of the above equation was established in[DKZ] (Ann. Prob. 47:1,2019). The case where $I=\mathbb{R}$ was left open. Theessential obstacle is caused by the explosion of the supremum norm of thesolution, $\sup_{x\in\mathbb{R}}|u(t,x)|=\infty$, making the usual truncationprocedure invalid. In this paper, we prove that there exists a unique globalsolution to the stochastic reaction-diffusion equation on the whole real line$\mathbb{R}$ with logarithmic nonlinearity. Because of the nature of thenonlinearity, to get the uniqueness, we are forced to work with the first ordermoment of the solutions on the space $C_{tem}(\mathbb{R})$ with a speciallydesigned norm $$\sup_{t\leq T, x\in\mathbb{R}}\left(|u(t,x)|e^{-\lambda|x|e^{\beta t}}\right),$$ where, unlike the usual norm in$C_{tem}(\mathbb{R})$, the exponent also depends on time $t$ in a particularway. Our approach depends heavily on the new, precise lower order momentestimates of the stochastic convolution and a new type of Gronwall'sinequalities we obtained, which are of interest on their own right.

Global well-posedness to stochastic reaction-diffusion equations on the  real line $\mathbb{R}$ with superlinear drifts driven by multiplicative  space-time white noise