Delta-shocks as limits of vanishing viscosity for a nonhomogeneous hyperbolic system

The Riemann problem for a 2\times 2 hyperbolic system of conservation laws with time-dependent damping is discussed. Making use of the variable substitution method, we construct six kinds of Riemann solutions including rarefaction wave, contact discontinuity, shock wave as well as delta-shock. Under suitable generalized Rankine–Hugoniot relation and entropy condition, both existence and uniqueness of delta-shock solution are established. Because of the appearing of the damping, the Riemann solutions are no longer self-similar. All wave curves are monotonic and have convexity-concavity. Moreover, by employing the vanishing viscosity method, we introduce a time-dependent viscous system to prove the stability of the solutions containing the delta-shocks.