Spatial decay solutions of the Boltzmann equation: Converse properties of long time limiting behavior

We prove some converse properties of long time limiting behavior (along the particle paths) of a class of spatial decay solutions of the Boltzmann equ ation. It is shown that different initial data f(0) determine different lon g time limit functions f(infinity)(x, v) = lim(t-->infinity) f(x + tv, v, t ), and for any given function F(x, v) which belongs to a function set, ther e exists a solution f such that f(infinity) = F. Existence of such spatial decay solutions are proven for the inverse power potentials with weak angul ar cut-off condition and for the initial data f(0) satisfying f(0)(x, v) le ss than or equal to C(1 + \x\(2) + \v\(2))(-k), or f(0)(x, v) less than or equal to C(1 + \x - v\(2))(-k), etc. For the soft potentials, the solutions may have "locally infinite particles," i.e., integral(R3) f(x, v, t)dv = i nfinity.