From Longman Dictionary of Contemporary Englishrun after somebody/something phrasal verb1 RUNto chase someone or something He ran after her, calling her name.2 informalSEX/HAVE SEX WITH to try to start a sexual relationship with someone He’s always running after younger women.3 spokenDO to do a lot of things for someone else as though you were their servant I can’t keep running after you all day! → run→ See Verb table
Examples from the Corpus
run after • That was why it was so odd that he was running after his wife.• Jonnie jumped up, tried to run after Jessica, but trapped in her narrow skirt, fell.• Finally a parson and a sexton get stuck, too, and have to run after Simpleton and his goose.• I had a twinge of hard joy as I ran after the car.• Philippa ran after them and peered down.• A crowd ran after them, jeering all the way.• Ryker vaulted the barrier and ran after them, slipping one hand into his jacket, touching the hilt of the knife.• There's a gap in the crowd and I dart through and run after them. 