Viscosity solutions and uniqueness for systems of inhomogeneous balance laws

This paper is concerned with the Cauchy problem $(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\overline{u}(x),$ for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear. Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\overline{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique "viscosity solution", defined for $t\in [0,T]$, depending continuously on the initial data.