  For high order (than two) in time operator-splitting methods applied to
dissipative systems, a folklore issue is the appearance of
negative-time/backward-in-time linear evolution operators such as backward
  heat operators interwoven with nonlinear evolutions. The stability of such
methods has remained an ensuing difficult open problem. In this work we
consider a fourth order operator splitting discretization for the Allen-Cahn
equation which is a prototypical high order splitting method with negative
time-stepping, i.e. backward in time integration for the linear parabolic part.
We introduce a new theoretical framework and prove uniform energy stability and
higher Sobolev stability. This is the first strong stability result for
negative time stepping operator-splitting methods.

Numerical Analysis;Numerical Analysis;Analysis of PDEs