1.
If you ~ something that is needed or needs dealing with, you find it or find out what is being done about it.
When I didn’t hear from the suppliers or receive a refund, I chased the matter up...
The authority can ~ the source of the pollution and demand that the owner clean it up.
PHRASAL VERB: V n P, V P n (not pron)
2.
If you chase someone up, you look for them and find them because you want them to do something or give you something.
...the story of a man who comes to Hollywood to ~ a client who has defaulted on a debt.
= track down
PHRASAL VERB: V P n (not pron), also V n P