Optimal decay of the 3D compressible Euler equations with general damping

In this paper, we study the large-time behaviors of global solutions for the Cauchy problem of the three-dimensional full compressible Euler equations with general damping. Under the initial data lie in [Formula: see text], we obtain the upper bounds of decay rates of the solution and its derivatives of any order by using Fourier analysis methods. Moreover, we prove the optimal decay rates in the sense that the lower bounds of decay rates are consistent with the upper ones when [Formula: see text]. Our methods can apply for a large class of systems of partial differential equations with lower order dissipation.