On existence of a global solution to the initial-boundary value problem for the Boltzmann equation

For the initial-boundary value problem
\begin{gather*} \frac{\partial f}{\partial t}+\biggl(v,\frac{\partial f}{\partial x}\biggr)=I(f,f), \quad (t,x,v)\in(0,T]\times G\times R_3^v, \\ f(t,x,v)|_{t=0}=f^0(x,v), \quad (x,v)\in G\times R_3^v, \\ f(t,x_{\partial G},v)=g(t,x_{\partial G},v), \quad (v,n_{\partial G})<0, \end{gather*}
the existence of a global solution in the space $C([0,T]$; $L^1(G\times R_3^v))$ $\forall\,T<\infty$ is proved under the conditions
\begin{gather*} f^0\in L^1\bigl(G\times\mathbf{R}_3^v\bigr), \quad f^0\ge0, \\ \int_{G\times\mathbf{R}_3^v}f^0(1+|v|^2+|{\ln f^0}|)\,dx\,dv<\infty, \quad |v|g\in C\bigl([0,T];\,L^1\bigl(\partial G\times\mathbf{R}_3^-\bigr)\bigr), \quad g\ge0, \\ \sup_{t\in[0,T]}\int_{\partial G\times\mathbf{R}_3^-}|v|g(1+|v|^2+|{\ln g}|)\,dx\,dv<\infty, \quad B\in L^1\bigl(S_2\times\mathbf{R}_3^v\bigr), \quad B\ge0, \end{gather*}
where $G$ is a bounded convex domain in $\mathbf{R}_3^x$ with boundary $\partial G$; $B$, the collision cross section; $S_2$, the unit sphere; and $\mathbf{R}_3^-=\bigl\{v\in\mathbf{R}_3^v:(v,n_{\partial G})<0\bigr\}$.