Riemann problem for a $2\times 2$ hyperbolic system with time-gradually-degenerate damping

Abstract This paper is focused on the Riemann problem for a $2\times 2$ 2 × 2 hyperbolic system of conservation laws with a time-gradually-degenerate damping. Two kinds of non-self-similar solutions involving the delta-shocks and vacuum are obtained using the variable substitution method. The generalized Rankine-Hugoniot relation and entropy condition are clarified for the delta-shock. Furthermore, the vanishing viscosity method proves the existence, uniqueness, and stability of non-self-similar solutions.