  Consider the stochastic evolution equation in a separable Hilbert space with
a nice multiplicative noise and a locally Dini continuous drift. We prove that
for any initial data the equation has a unique (possibly explosive) mild
solution. Under a reasonable condition ensuring the non-explosion of the
solution, the strong Feller property of the associated Markov semigroup is
proved. Gradient estimates and log-Harnack inequalities are derived for the
associated semigroup under certain global conditions, which are new even in
finite-dimensions.
