badmouth vs asperse

badmouth and asperse both are verbs.

Word	Noun	Adjective	Verb	Adverb
badmouth	No	No	Yes	No
asperse	No	No	Yes	No

As verbs, asperse is a hypernym of badmouth; that is, asperse is a word with a broader meaning than badmouth:

badmouth: speak unfavorably about
asperse: charge falsely or with malicious intent; attack the good name and reputation of someone

Other hypernyms of badmouth include besmirch, calumniate, defame, denigrate, slander, smear, smirch, sully.

badmouth (verb)	asperse (verb)
speak unfavorably about	charge falsely or with malicious intent; attack the good name and reputation of someone