  In this paper, we proceed to study the nonlocal diffusion problem proposed by
Li and Wang [8], where the left boundary is fixed, while the right boundary is
a nonlocal free boundary. We first give some accurate estimates on the longtime
behavior by constructing lower solutions, and then investigate the limiting
profiles of this problem when the expanding coefficient of free boundary
converges to $0$ and $\yy$, respectively. At last, we focus on two important
kinds of kernel functions, one of which is compactly supported and the other
behaves like $|x|^{-\gamma}$ with $\gamma\in(1,2]$ near infinity. With the help
of some upper and lower solutions, we obtain some sharp estimates on the
longtime behavior and rate of accelerated spreading.
