  As the most significant difference from parabolic equations, long-time or
short-time behavior of solutions to time-fractional evolution equations is
dominated by the fractional orders, whose unique determination has been
frequently investigated in literature. Unlike all the existing results, in this
article we prove the uniqueness of orders and parameters (up to a multiplier
for the latter) only by principal terms of asymptotic expansions of solutions
near $t=0$ at a single spatial point. Moreover, we discover special conditions
on unknown initial values or source terms for the coincidence of observation
data. As a byproduct, we can even conclude the uniqueness for initial values or
source terms by the same data. The proof relies on the asymptotic expansion
after taking the Laplace transform and the completeness of generalized
eigenfunctions.
