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1. To look for someone or something in a variety of places. I went to five different hardware stores and was finally able to chase up that part for the heater.
2. To pursue someone or something (often an animal) and cause it to run up something (such as a tree). A noun or pronoun can be used between "chase" and "up." I had to call for help when my dog chased the neighbor's cat up a tree.
chase someone or something up something
to drive someone or an animal up something, such as a tree, a hill, a cliff, etc. The ranchers chased the mountain lion up a tree. The bull chased the rodeo clown up into the stands.
chase someone or something up
to seek someone or something out; to look high and low for someone or something. I will chase Tom up for you. I will try to chase up a buyer for your car.