Asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem (Q2371877)

The author studies the initial value problem to the system \(v_t-u_x=0,\, u_t-f(v)_x=u_{xx}\) subjected to discontinuous initial data, which is related to the phase transition problem. He proves the existence of the solutions \((v,u)\) and their asymptotic decay to a piecewise constant state \((\overline{v}_{\pm},\overline{u}),\) where two phases coexist. He shoes, that Rankine--Hugoniot condition across the line \(x=0,\) which implies the continuity of \(u,\) is satisfied. Both hyperbolic elliptic mixed type and hyperbolic type of the system are discussed.