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We consider the initial value problem of Fokker-Planck-Boltzmann（FPB） equations:where f = f（x,v,t）, （x,v,t）∈R3×R3×R+, f is an unknown function.In this paper, we are mainly concerned with the global existence and uniqueness of the strong solution of FPB equation with initial data near M0 （absolute Maxwellian） and it’s long time behavier.For the spatial inhomogeneous FPB equation, we prove that as the initial data is a small perturtation of M0 and the viscous coefficient is small enough but fixed, the Cauchy problem （1） has a unique strong solution globally in time. Moreover, if the Lv2（Lx1） norm of f0 - M0 is finite,we obtain the algebra time decay rate of the solution to M0.For the spatial homogeneous FPB equation, we prove that as the initial data is a small perturtation of M0 and the viscous coefficient is small but fixed, the Cauchy problem has a unique strong solution globally in time, which tends to M0 exponentially in time.
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