The most common and most likely answer to this crossword clue is the 5 letter word INCUR
I've seen this crossword clue in The Times
I
N
C
U
R
More clues leading to the result INCUR
- Suffer at home with vicious mutt
- Come by home with vicious dog
- Suffer with one new dog that’s aggressive
- At home, aggressive dog makes one suffer
- Risk including what’s central to source
- Run through check endlessly
- Expose oneself to copper wearing hip designer’s latest
- Suffer at home with scoundrel
- Be liable for where dog’s dinner might end up?
- Run into S American once leaving a historic city
Buddy explains!

The straight part
We know that "Run up!" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case run up! = incur.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 INCUR mean?
INCUR verb- To bring upon oneself or expose oneself to, especially something inconvenient, harmful, or onerous; to become liable or subject to
- To enter or pass into
- To fall within a period or scope; to occur; to run into danger