2021 Volume 64 Issue 1 Pages 49-73
We investigate the asymptotic behavior in time of solutions to the Cauchy problem for the one-dimensional dissipative wave equation where the far field states are prescribed. In particular, we study the case where the flux function is convex but linearly degenerate on some interval. When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward the linear combination of the rarefaction waves and viscous contact wave as time goes to infinity. The proof is given by a technical energy method under the sub-characteristic condition. We also show that the similar arguments are applicable to the initial-boundary value problem on the half space.