Balance Laws with Integrable Unbounded Sources

We consider the Cauchy problem for an n× n strictly hyperbolic system of balance laws uₜ+f(u)ₓ=g(x,u), x, $t>0$, \|g(x,·)\|C²̃(x)¹, endowed with the initial data u(0,.)=uₒ¹(R;Rⁿ). Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., |λᵢ(u)|≥ c>0 for all i∈\1,,n\. Assuming that the L¹ norms of \|g(x,·)\|C¹ and \|uₒ\|BV(R) are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D. Amadori, L. Gosse, and G. Guerra, Arch. Ration. Mech. Anal., 162 (2002), pp. 327–366] to unbounded (in L^∞) 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.