In this work we address the problem of the existence of quasi steady-states
in the solutions of the linear, one-dimensional Boltzmann equation in regi
me of runaway. It is known that runaway occurs when the collision frequency
is integrable, that is, when collisions are not sufficiently strong to cou
nterbalance the effect of the external field. On the other hand, when the c
ollision frequency is constant, then the time dependent solution relaxes to
the solution of the stationary problem and the average velocity approaches
a constant value asymptotically in time. In this paper we shall show analy
tically that, if the collision frequency is given by an integrable function
which is also close to a constant, then the solution and the average veloc
ity for this model approximately follow the solution and the average veloci
ty of the model with constant collision frequency for some time before fina
lly breaking off to form a runaway wave. Thus, even in a runaway regime, th
ere is a period of time when the average velocity is close to a constant: w
e call this a quasi steady-state or a plateau, This result is confirmed by
a series of numerical examples performed for BGK models with collision freq
uences having compact support. As a by-product, we have found, so far heuri
stically, that there is a transient state between the quasi steady-state an
d the asymptotic linear state where the evolution of the average velocity i
s quadratic in time.