For the nonstationary Boltzmann equation $$ \frac{\partial F}{\partial t}+\xi_\alpha\frac{\partial F}{\partial x_\alpha}=Q(F,F),\qquad t>0,\quad\xi\in R^3,\quad x\in\Omega\subset R^3, $$ one proves the unique global solvability of the Cauchy problem under nondifferentiable initial data and the unique global solvability of initial-boundary-value problems with homogeneous boundary conditions; it is shown that the solutions of the initial-boundary-value problems decay exponentially as $t\to\infty$.