Seve, Chase = pursue 1) Chase up = to pursue (a matter, person, etc.) vigorously with a specific purpose, esp. after an earlier unsatisfactory response- colloq. (OED) When I didn't hear from the suppliers or receive a refund, I chased the matter up. 2)Chase up =to make efforts to find or obtain quickly. (OED) It is the story of a man who comes to Hollywood to CHASE UP his twin brother whom he hasn't seen since childhood. The second definition of "chase up" means the same as "chase down". (English can be an odd language) Chase down =to try hard to find or get something As soon as we CHASE DOWN further information, we will bring it to you.