On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping { ∂ t ρ + d i v ( ρ u ) = 0 , ∂ t ( ρ u ) + d i v ( ρ u ⊗ u + p I d ) = − α ( t ) ρ u , ρ ( 0 , x ) = ρ ¯ + ε ρ 0 ( x ) , u ( 0 , x ) = ε u 0 ( x ) , where x = ( x 1 , ⋯ , x d ) ∈ R d ( d = 2, 3), the frictional coefficient is </jats: