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1.
The fight song sung at Chicago Bears home games by fans everytime they score. Also a term used to acknowledge other Bears fans.
Bear down, Chicago Bears! Make every play clear the way to victory!
Bear down, Chicago Bears! Put up a fight with a might so fearlessly!
We'll never forget the way you thrilled the nation with your T formation.
Bear down, Chicago Bears, and let them know why you're wearing the crown.
You're the pride and joy of Illinois. Chicago Bears, bear down!

Guy1:<walks past another Bear fan> BEAR DOWN!
Guy2: DA BEARSS! <High Five>
by MCK83 May 02, 2008
 
2.
The school motto of the University of Arizona. In 1926, after Arizona Wildcats star football quarterback John Salmon was involved in a horrific car accident, he was hospitalized and visited daily by his coach Pop McKale. On October 18, 1926, Salmon spoke his last words to his coach before dying, "Tell the team to bear down." Coach McKale told the Wildcats team as Salmon wished before a big game, and Arizona wound up winning. It became the school's motto shortly afterwards.

The phrase itself basically means to be strong and smart. It has nothing to do with any large carnivorous animal.
Dude 1: "Bear down!"

Dude 2: "Wait, why would Wildcats bear down? Wildcats aren't bears! They stole it from Chicago!"

Dude 1: "Probably because the phrase has absolutely nothing to do with bears in the first place, it means to exert strength and intelligence. And Arizona has been using it since 1926. Chicago didn't start using until nearly twenty years later!"

Dude 2: "Oh........."
by 98712342634234 March 29, 2014
 
3.
The act of a big, hairy gay man, mounting his "Cub". i.e., his bottom.
After some drinks and loosening up, Gary was ready to bear down on Scott.
by The E-Radicator January 19, 2011
 
4.
When you want a Chicago Bears fan to get on their knees (down) & give you a blow job.
Bear down, this dick isn't going to blow itself.
by Yourmomiseasy June 17, 2011