Balance laws with integrable unbounded sources

We consider the Cauchy problem for an n x n strictly hyperbolic system of balance laws u(t) + f(u)(x) = g(x, u), x is an element of R, t > 0, parallel to g(x, .)parallel to(C2) <= (M) over tilde (x) is an element of L-1, endowed with the initial data u(0, .) = u(o) is an element of L-1 boolean AND BV(R; R-n). Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., vertical bar lambda(i)(u)vertical bar >= c > 0 for all i is an element of {1, ..., n}. Assuming that the L-1 norms of parallel to g(x, .)parallel to(C1) and parallel to u(o)parallel to(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-infinity) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showin...