back down vs cave in

back down

verb
  • To withdraw from a commitment or position. 

  • To take a less aggressive position in a conflict than one previously has or has planned to. 

cave in

verb
  • To relent; to grant approval against one's initial will. 

  • To collapse inward or downward. 

noun
  • The act of relenting. 

  • The location where something has caved in. 

  • The act of something collapsing or caving in. 

How often have the words back down and cave in occurred in a corpus of books? (source: Google Ngram Viewer )