In this paper, we consider the Cauchy problem for linear first-order partial differential equations with discontinuous coefficients. We show that, under suitable assumptions, it has a unique continuous solution. Moreover, this solution is stable under perturbation of the coefficients. We also show that this solution can be expressed explicitly by integrating along generalized characteristics.
"Generalized solutions of linear partial differential equations with discontinuous coefficients." Differential Integral Equations 17 (5-6) 653 - 668, 2004.