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 $L^\infty$ norm. 
Long Fan, Qingqing Liu, Changjiang Zhu
math.AP