  For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a
constant $\ll>0$ is called a lower bound of the ultra-exponential convergence
rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that
$$ \sup_{\mu(f^2)\le 1}\|P_tf-\mu(f)\|_\infty \le C \e^{-\ll t},\ \ t\ge 1.$$
By using the coupling by change of measure in the line of [F.-Y. Wang, Ann.
Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential
convergence rate are derived for a class of non-linear monotone stochastic
partial differential equations. The main result is illustrated by the
stochastic porous medium equation and the stochastic $p$-Laplace equation
respectively. Finally, the $V$-uniformly exponential convergence is
investigated for stochastic fast-diffusion equations.
