Convergence Rates to Stationary Solutions of a Gas-liquid Model with External Forces and Vacuum

In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and $\gamma >1$, if $\theta \in (0,\frac{\gamma }{2}]\cap (0,\gamma -1]\cap (0,1-\alpha \gamma ]$, we prove the weak solution $(cQ(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if $\theta \in (0,\frac{\gamma }{2}]\cap (0,\gamma -1)\cap (0,1-\alpha \gamma ]$, following the same idea in \cite{Fang-Zhang4}, we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^{\mathrm{\infty }}$ norm.