They were absorbed into Alexander's empire when he conquered the Persian Empire. When that empire was broken up into the Hellenistic kingdoms by his successors, they were either absorbed into those kingdoms or remained as independent city-states until they were absorbed into the expanding Roman Empire which took over the kingdoms and city-states/
Persian king Cambyses brought Egypt into the Persian Empire.
Persia crushed the Ionian Revolt in Asia Minor.
Which Persian War? Persia fought the Greek city-states over 2,000 years ago; is your question regarding that Persian War?
Some of the Greek city-states joined with Persia, the southern cities opposed the Persian attempt to absorb them into the Persian empire.
Alexander's conquest.
Is this a question about civil war in Persia.
Persia incorporated the Babylonian Empire into the Persian Empire.
Persia (once was Persia).
persian strengths
Persia did not conquer Greece. Persia attempted to conquer Greece but the invasion was repelled. Earlier, Greeks had fought against Persia in the Ionian Revolt, and the Persian invasion was intended as retribution for the violation of Persian temples.
It was long drawn out over 50 years. The first phase was Persia subduing rebellions by Greek cities within the Persian empire. The next phase was a Persian invasion of mainland Greece to attempt to bring it within the empire. This failing, warfare became more sporadic, but going in favour of a Greek alliance This brought about a peace, with Persia accepting the independence of the Greek cities in Asia Minor. Athens, the leader of this alliance, did not accept the independence of these 'liberated' cities, coercing them to stay in what Athens turned into an empire of its own.
At Marathon and Salamis, the Greek cities defeated the Persian forces. At Thermopylai the Persian forces defeated the Greek cities.