Delta shock interactions and stability of the Riemann solutions in generalized Chaplygin gas with time-dependent general source term

This paper presents the Riemann problem and wave interactions for the one-dimensional generalized Chaplygin gas equations with a nonlinear time-dependent source term. We rigorously construct the Riemann solutions for this nonhomogeneous hyperbolic system and show that, for a class of initial data, the solutions develop singular measures in the form of δ-shock waves. For such solutions, we derive the generalized Rankine–Hugoniot conditions and obtain explicit formulas for the propagation speed, trajectory, and strength of the δ-shock through a suitable δ-entropy framework. We further provide a detailed analysis of wave interactions involving δ-shocks by introducing three-state piecewise constant perturbations, which enables the construction of global-in-time solutions. By performing a vanishing perturbation limit as ɛ → 0, we establish the stability of the Riemann solutions and rigorously characterize their asymptotic behavior. These results extend the theory of δ-shocks to nonhomogeneous systems with time-dependent source terms and offer new insights into the interplay between source effects and singular wave dynamics.