(transitive) To lessen the intensity of, to mitigate or relieve (hunger, emotion, pain, etc.).
1864 November 21, Abraham Lincoln (signed) or John Hay, letter to Mrs. Bixby in Boston
I pray that our Heavenly Father may assuage the anguish of your bereavement, and leave you only the cherished memory of the loved and lost.
(transitive) To pacify or soothe (someone).
(intransitive, obsolete) To calm down, become less violent (of passion, hunger etc.); to subside, to abate.