From Longman Dictionary of Contemporary Englishpeter out phrasal verbSTOP HAPPENINGto gradually become smaller, less, weaker etc and then come to an end The road became narrower and eventually petered out. Public interest in the environment is in danger of petering out. → peter→ See Verb table
Examples from the Corpus
peter out• By midday the rain had petered out.• The protest campaign petered out after a few weeks.• The road petered out into a muddy track. 