Vanishing viscosity limit for a system of balance laws with general type initial data arising from 1D Saint-Venant model

We study the vanishing viscosity limit of a non-strictly hyperbolic system of balance laws known as the 1D Saint-Venant model whose solution admits δ-waves. The balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of Olenik [Usp Math. Nauk. 12(3), 3–73 (1957) (in Russian)]. We also studied the large time behavior when the viscosity parameter is positive. When the balance term depends only on the time variable, we obtained an explicit formula for the solution.