On the global existence for the Kac model with external force

The nonlinear Boltzmann Kac’s model under the influence of a external force field arising from a potential ϕ=ϕ(x)ϕ=ϕ(x) is considered, and the global existence and uniqueness for the general solution to the initial value problem is proved. The result is proved for small initial data in the case potential bounded from below. As an application, some results concerning the harmonic oscillator are obtained.