报告题目 (Title)：Global well-posedness to stochastic reaction-diffusion equations on the real line with superlinear drifts driven by multiplicative space-time white noise （可乘时空白噪声驱动的超线性漂移实线上随机反应扩散方程的全局适定性）

报告人 (Speaker)：张土生 教授（中国科学技术大学）

报告时间 (Time)：2022年9月7日 (周三) 9:00

报告地点 (Place)：腾讯会议（会议号：462-567-695 无密码）

邀请人(Inviter)：张阳春

主办部门：理学院数学系

报告摘要：Consider 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.