  We study the effect of the elastic scattering on the non-thermal WIMP, which
is produced by direct decay of heavy particles at the end of reheating. The
non-thermal WIMP becomes important when the reheating temperature is well below
the freeze-out temperature. Usually, two limiting cases have been considered.
One is that the produced high energetic dark matter particles are quickly
thermalized due to the elastic scattering with background radiations. The
corresponding relic abundance is determined by the thermally averaged
annihilation cross-section at the reheating temperature. The other one is that
the initial abundance is too small for the dark matter to annihilate so that
the final relic is determined by the initial amount itself. We study the
regions between these two limits, and show that the relic density depends not
only on the annihilation rate, but also on the elastic scattering rate.
Especially, the relic abundance of the p-wave annihilating dark matter
crucially relies on the elastic scattering rate because the annihilation
cross-section is sensitive to the dark matter velocity. We categorize the
parameter space into several regions where each region has distinctive
mechanism for determining the relic abundance of the dark matter at the present
Universe. The consequence on the (in)direct detection is also studied.
