  The parabolic Anderson model is the Cauchy problem for the heat equation with
a random potential. We consider this model in a setting which is continuous in
time and discrete in space, and focus on time-constant, independent and
identically distributed potentials with polynomial tails at infinity. We are
concerned with the long-term temporal dynamics of this system. Our main result
is that the periods, in which the profile of the solutions remains nearly
constant, are increasing linearly over time, a phenomenon known as ageing. We
describe this phenomenon in the weak sense, by looking at the asymptotic
probability of a change in a given time window, and in the strong sense, by
identifying the almost sure upper envelope for the process of the time
remaining until the next change of profile. We also prove functional scaling
limit theorems for profile and growth rate of the solution of the parabolic
Anderson model.
