Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws

We consider the Cauchy problem for $n\times n$ strictly hyperbolic systems of nonresonant balance laws $$ \left\{\begin{array}{c} u_t+f(u)_x=g(x,u), \qquad x \in \reali, t>0\\ u(0,.)=u_o \in \L1 \cap \BV(\reali; \reali^n), \\ | \la_i(u)| \geq c > 0 \mbox{ for all } i\in \{1,\ldots,n\}, \\ |g(.,u)|+\norma{\nabla_u g(.,u)}\leq \om \in \L1\cap\L\infty(\reali), \\ \end{array}\right. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that $\|\om\|_{\L1(\reali)}$ and $\|u_o\|_{\BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.