The most common and most likely answer to this crossword clue is the 10 letter word EXHILARATE
I've seen this crossword clue in The Guardian
E
X
H
I
L
A
R
A
T
E
More clues leading to the result EXHILARATE
- Axe the liar, needing change — cheer up, bigly
- Cheer short girl that’s half submerged in a river
- I hear latex is kinky, providing thrill
- Cheer when greeting the French deserter found in river
- Thrill created by old lover, a real hit when reunited
- Cheer up exile welcoming, variously, Henry and a rat
- Reapplication of heat and I relax in delight
- Cheer up, greeting cricketer with tea in the river
- Phil into sex and karate, initially out to thrill
- Thrill in the midst of Sussex hilltop – at a price!
Buddy explains!

The straight part
We know that "cheer up, bigly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cheer up, bigly = exhilarate.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EXHILARATE mean?
EXHILARATE verb- To cheer, to cheer up, to gladden, to make happy.
- To excite, to thrill.