Global existence and large time behavior for the system of compressible adiabatic flow through porous media in R³

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The system of compressible adiabatic flow through porous media is considered in R³ 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 L²-rate (1+t)⁻³/⁴ as the Navier-Stokes equations without heat conductivity, but the velocity of the system decays at the L²-rate (1+t)⁻⁵/⁴, which is faster than the L²-rate (1+t)⁻³/⁴ for the Navier-Stokes equations without heat conductivity [3].