  We describe the large-time moment asymptotics for the parabolic Anderson
model where the speed of the diffusion is coupled with time, inducing an
acceleration or deceleration. We find a lower critical scale, below which the
mass flow gets stuck. On this scale, a new interesting variational problem
arises in the description of the asymptotics. Furthermore, we find an upper
critical scale above which the potential enters the asymptotics only via some
average, but not via its extreme values. We make out altogether five phases,
three of which can be described by results that are qualitatively similar to
those from the constant-speed parabolic Anderson model in earlier work by
various authors. Our proofs consist of adaptations and refinements of their
methods, as well as a variational convergence method borrowed from finite
elements theory.
