  In this paper we present a formal analysis of the long-time asymptotics of a
particular class of solutions of the Boltzmann equation, known as homoenergetic
solutions, which have the form $f\left( x,v,t\right)=g\left( v-L\left( t\right)
x,t\right)$ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ with the matrix
$A$ describing a shear flow or a dilatation or a combination of both. We began
this study in \cite{JNV1}. Homoenergetic solutions satisfy an
integro-differential equation which contains, in addition to the classical
Boltzmann collision operator, a linear hyperbolic term. In \cite{JNV1} it has
been proved rigorously the existence of self-similar solutions which describe
the change of the average energy of the particles of the system in the case in
which there is a balance between the hyperbolic and the collision term. In this
paper we focus in homoenergetic solutions for which the collision term is much
larger than the hyperbolic term (collision-dominated behavior). In this case
the long time asymptotics for the distribution of velocities is given by a time
dependent Maxwellian distribution with changing temperature.
