SQL Server 2012 Page Life Expectancy resets to 0 after about 50 days
I've seen this behaviour on a client site running SQL2012 SP1.
The specifics here were NUMA, and PLE demonstrating a 'sawtooth' pattern but on an hourly cycle.

A couple of threads on SQLServerCentral discussed around this:

http://www.sqlservercentral.com/Forums/Topic1415833-2799-1.aspx
http://www.sqlservercentral.com/Forums/Topic1424826-2799-1.aspx

the end result being that applying SP1 CU4 seemed to fix the problem.

CU4 contains the innocent looking fix An update is available for SQL Server 2012 Memory Management KB2845380

Worth a try?