Ergodicity of reflected stochastic reaction-diffusion equations driven by space-time white noise

We consider the reflected stochastic reaction-diffusion equation on $[0,1]$: \begin{align*}
\left\{
\begin{aligned}
d u(t,x) &=\frac{1}{2}\partial_{xx} u(t,x)dt +b(u(t,x))dt + \sigma(u(t,x)) W(dt,dx)+L(dt,dx),\\
u(t,x)&\geq 0, \quad t\geq 0, \ x\in [0,1],\\
u(0,x)&=u_0(x)\geq 0, \quad x\in [0,1],\\
u(t,0) &= u(t,1) = 0, \quad \forall\ t\geq 0,
\end{aligned}
\right. \end{align*} where the initial value $u_0$ is non-negative on $[0,1]$ satisfying $u_0(0)=u_0(1)=0$, and $ W(dt,dx)$ is a space-time white noise. The $L$ in the equation is a random measure on $[0,\infty)\times(0,1)$, which is a part of the solution pair $(u, L)$.
In this paper, we establish the existence and uniqueness of invariant measures, as well as exponential mixing for the reflected stochastic reaction diffusion equation under the dissipative condition $$(b(x)-b(y))(x-y)\leq -\alpha (x-y)^2,$$ which include the coefficients having polynomial, even exponential growth. The big obstacle of utilizing the dissipative condition is the lack of the Itô formula/energy equality for such equations. To circumvent the problem, we use the newly found method in our paper (arXiv:2606.26619, 2026) to fully exploit comparison principles of reflected stochastic reaction-diffusion equation.