Standard Time-to-Live (TTL) cache management prescribes the storage of entire files, or possibly fractions thereof, for a given amount of time after a request. As a generalization of this approach, this work proposes the storage of a time-varying, diminishing, fraction of a requested file. Accordingly, the cache progressively evicts parts of the file over an interval of time following a request. The strategy, which is referred to as soft-TTL, is justified by the fact that traffic traces are often characterized by arrival processes that display a decreasing, but non-negligible, probability of observing a request as the time elapsed since the last request increases. An optimization-based analysis of soft-TTL is presented, demonstrating the important role played by the hazard function of the inter-arrival request process, which measures the likelihood of observing a request as a function of the time since the most recent request.