  In this paper, we consider a particular class of solutions to the Boltzmann
equation which are referred to as homoenergetic solutions. They describe the
dynamics of a dilute gas due to collisions and the action of either a shear, a
dilation or a combination of both. We prove that solutions with initially high
temperature remain close and converge to a Maxwellian distribution with
temperature going to infinity. Furthermore, we give precise asymptotic formulas
for the temperature. This local stability result is a consequence of a dominant
shear and the homogeneity $ \gamma>0 $ of the collision operator with respect
to relative velocities. The proof relies on an ansatz which is motivated by a
Hilbert-type expansion. We consider both non-cutoff and cutoff kernels.
