Balance laws with integrable unbounded sources

We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$\{arrayc{u}_t+f(u{)}_x=g(x,u),x\in \mathbb{R},t>0u(0,\mathrm{.})=u_o\in L^1\cap BV(\mathbb{R};{\mathbb{R}}^n),\mathrm{\mid }{\lambda }_i(u)\mathrm{\mid }\ge c>0foralli\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},n\},\mathrm{\parallel }g(x,\cdot ){\mathrm{\parallel }}_{{\mathbf{C}}^2}\le \tilde{M}(x)\in L1,array\mathrm{.}$$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the ${\mathbf{L}}^1$ norm of $\mathrm{\parallel }g(x,\cdot ){\mathrm{\parallel }}_{{\mathbf{C}}^1}$ and $\mathrm{\parallel }{u}_o{\mathrm{\parallel }}_{BV(\text{reali})}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^{\mathrm{\infty }}$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.