run after somebody/something phrasal verb (see also run)

1. to chase someone or something:
He ran after her, calling her name.

2. informal to try to start a sexual relationship with someone:
He’s always running after younger women.

3. spoken to do a lot of things for someone else as though you were their servant:
I can’t keep running after you all day!

[TahlilGaran] Dictionary of Contemporary English ▲

run after
run after (someone)
to try to get the attention or love of someone.
He's about to make a fool of himself, running after a married woman.

Etymology: based on the literal meaning of run after someone or something (= to chase someone or something)

[TahlilGaran] English Idioms Dictionary ▲

run after
v.
1. To try to find; look for; hunt.
The Dramatic Club has to run all over town after things for setting the stage when it puts on a play.
2. informal To seek the company of; chase.
Some boys spend a lot of time and money running after girls.

[TahlilGaran] English Idioms Dictionary ▲