»ã±¨±êÌâ (Title)£ºGlobal well-posedness to stochastic reaction-diffusion equations on the real line with superlinear drifts driven by multiplicative space-time white noise £¨¿É³Ëʱ¿ÕȱÔëÉùÇý¶¯µÄ³¬ÏßÐÔƯÒÆʵÏßÉÏËæ»ú·´Ó³À©É¢·½³ÌµÄÈ«¾ÖÊʶ¨ÐÔ£©

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»ã±¨¹¦·ò (Time)£º2022Äê9ÔÂ7ÈÕ (ÖÜÈý) 9:00

»ã±¨µØÖ· (Place)£ºÌÚѶ»áÒ飨»áÒéºÅ£º462-567-695 ÎÞÃÜÂ룩

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»ã±¨ÌáÒª£ºConsider the stochastic reaction-diffusion equation with logarithmic nonlinearity driven by space-time white noise:

\begin{numcases}{} du(t,x) = \frac{1}{2}\Delta u(t,x) dt+ b(u(t,x)) dt \nonumber\\

~~~~~~~~~~~~~ + \sigma(u(t,x)) W(dt,dx), \ t>0, x\in I , \nonumber\\

\label{1.a} u(0,x)=u_0(x), \quad x\in I .\nonumber

\end{numcases}

When $I$ is a compact interval, say $I=[0,1]$, the well-posedness of the above equation was established in \cite{DKZ} (Ann. Prob. 47:1,2019).The case where $I=\bR$ was left open. The essential obstacle is caused by the explosion of the superum norm of the solution, $\sup_{x\in\bR}|u(t,x)|=\infty$, making the usual trancation procedure invalid. In this paper, we prove that there exists a unique global solution to the stochastic reaction-diffusion equation on the whole real line $\mathbb{bR}$ with logarithmic nonlinearity. Because of the nonlinearity, to get the uniqueness, we are forced to work with the first order moment of the solutions on the space $C_{tem}(\bR)$.

Our approach depends heavily on the new, precise lower order moment estimates of the stochastic convolution and a new type of Gronwall's inqualities we obtained, which are of interest on their own right.