expression
never ceasing; never ending; endless
Fixed expression meaning that something (often negative events, visitors, or problems) continues without stopping. Used in formal and written contexts.
交通事故が後を絶たず、対策が求められている。
Traffic accidents never cease, and countermeasures are being called for.
彼の店には客が後を絶たず、いつも賑わっている。
His shop has an endless stream of customers and is always bustling.
絶えない is a more general adjective meaning 'unceasing' or 'constant', while 後を絶たず is a fixed phrase specifically meaning that something continues without interruption, often with a nuance of being problematic or noteworthy.
A fixed expression composed of 後 (あと, 'after' or 'trace') + を (object marker) + 絶たず (たたず, negative continuative form of 絶つ 'to cut off'). Literally 'without cutting off the trace', it came to mean that something continues without ceasing.