  The open-charm strong decays of higher charmonium states up to the mass of
the $6P$ multiplet are systematically studied in the $^3P_0$ model. The wave
functions of the initial charmonium states are calculated in the linear
potential (LP) and screened potential (SP) quark model. The decay widths for
most of the well-established charmonium states above the open-charm thresholds
can be reasonably described. By comparing our quark model calculations with the
experimental observations we also discuss the nature of some of the newly
observed charmonium-like states. It is found that (i) the $\psi(4415)$ may
favor the $\psi(4S)$ or $\psi_1(3D)$ assignment. There may exist two highly
overlapping vector charmonium states around 4.4 GeV; (ii) In the LP model the
$J^{PC}=1^{--}$ $Y(4660)$ resonance and the $J^{PC}=0^{++}$ $X(4500)$ resonance
may be assigned as the $\psi(5S)$ and $\chi_{c0}(4P)$, respectively; (iii) The
newly observed state $X^*(3860)$ can be assigned as the $\chi_{c0}(2P)$ state
with a narrow width of about $30$ MeV; (iv) It seems to be difficult to
accommodate the $X(4140)$ and $X(4274)$ states in the same potential model as
excited $\chi_{c1}$ states. (v) The $X(3940)$ resonance can be assigned as the
$\eta_c(3S)$ state; (vi) The vector charmonium-like states $Y(4230/4260,4360)$
and scalar $X(4700)$ cannot be described by any conventional charmonium states
self-consistently in our model.
