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to run after
[phrase form: run]
01
to follow someone or something in an attempt to catch them
Example
The police had to run after the suspect to apprehend him.
The children love to run after the ice cream truck when it comes through the neighborhood.
02
to actively try to gain someone's love or affection
Example
John has been running after Sarah for months, hoping she'll notice his feelings.
Despite numerous rejections, he continues to run after her, believing she will eventually say yes.