Global existence and large time behavior for the system of compressible adiabatic flow through porous media in ${\mathbb{R}}^3$

The system of compressible adiabatic flow through porous media is considered in ${\mathbb{R}}^3$ in the present paper. The global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium. We also show that the pressure of the system converges to its equilibrium state at the same $L^2$-rate $(1+t{)}^{-3\mathrm{/}4}$ as the Navier-Stokes equations without heat conductivity, but the velocity of the system decays at the $L^2$-rate $(1+t{)}^{-5\mathrm{/}4}$, which is faster than the $L^2$-rate $(1+t{)}^{-3\mathrm{/}4}$ for the Navier-Stokes equations without heat conductivity [3].