How does rainbow stay for such long time even it is caused due to raindrops? Rainbow forms when light rays pass through a raindrop (usually, it can be a watet droplet also) and it undergoes several phenomenon like refraction , reflection , dispersion , But after the rainbow is formed it should not stay for much long as the rain drops are falling downwards so their position must change . And the rainbow should disappear .But this doesn't happens in reality . Why is it so ?

 A: 
But after the rainbow is formed it should not stay for much long as the rain drops are falling downwards so their position must change . 

As discussed in the comments, and explained in the wiki article , the raindrops replace each other as they are falling and your eyes see a continuity similar to the one when watching movies and television.

And the rainbow should disappear

If it were only raindrops that could form a rainbow, i.e drops of some milimeters, yes it should disappear. But also tiny droplets  hovering after the main rain drops, create rainbows. See this over a water fountain, where the mist is enough on the left.

After the rain, and depending on the weather conditions, mist can persist:
The persistence of water vapor in the air depends on updrafts of warm air.  As warm air rises, it carries humidity that condenses into vapor droplets when it hits higher layers of colder air.  Once condensed, the vapor tends to fall.  But continuously rising warm air replenishes the fallen droplets, so the rainbow phenomenon persists as sunlight is refracted and reflected through new droplets that replace the old.  When warm updrafts cease, so that no further humidity is carried upward to condense to droplets, the rainbow will dissipate and disappear.
