The authors consider two classes of free boundary value problems for a simplified one-dimensional liquid-gas two-phase model \[ \begin{aligned} &\partial_t n+\partial_x(nu)=0,\\ &\partial_t m+\partial_x(mu)=0,\\ &\partial_t (mu)+\partial_x(mu^2+P)=\partial_x(\varepsilon\partial_xu), \end{aligned} \] with \[ \begin{aligned} &P(n,m)=\left(\frac{n}{\rho_l-m}\right)^\gamma,\;\gamma>1,\\ &\varepsilon=\varepsilon(n,m)=\frac{n^\beta}{(\rho_l-m)^{\beta+1}},\;\beta>0, \end{aligned} \] where \(\rho_l>0\) is a constant. The authors obtain the asymptotic behavior: \[ \lim_{t\to\infty}\sup_{x\in[0,1]}(n(x,t),m(x,t))=0, \] together with some algebraic decay rates of mass functions \(n(x,t),m(x,t)\) when the initial masses are assumed to be connected to vacuum both discontinuously and continuously.