Initial Value Problem Of The Fokker-planck-boltzmann Equation

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 L_v²(L_x¹) 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.