Quasi steady-state solutions of kinetic equations in runaway regime

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 regime of runaway. It is known that runaway occurs when the collision frequency is integrable, that is, when collisions are not sufficiently strong to counterbalance the effect of the external field. On the other hand, when the collision 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 analytically that, if the collision frequency is given by an integrable function which is also close to a constant, then the solution and the average velocity for this model approximately follow the solution and the average velocity of the model with constant collision frequency for some time before finally breaking off to form a runaway wave. Thus, even in a runaway regime, there is a period of time when the average velocity is close to a constant: we 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 frequences having compact support. As a by-product, we have found, so far heuristically, that there is a transient state between the quasi steady-state and the asymptotic linear state where the evolution of the average velocity is quadratic in time.