Global existence and blowup of smooth solutions of 3-D potential equations with time-dependent damping

In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ ∂_tρ+div(ρ u)=0, ∂_t(ρ u)+div(ρ u⊗ u+p\,I₃)=-\,{μ}{(1+t)λ}\,ρ u, ρ(0,x)= ρ+ερ_0(x), u(0,x)=ε u_0(x), $$ where $x∈ R^3$, $μ>0$, $λ≥ 0$, and $ρ>0$ are constants, $ρ_0,\, u_0∈ C_0∞( R^3)$, $(ρ_0, u_0)≡ 0$, $ρ(0,·)>0$, and $ε>0$ is sufficiently small. For $0≤λ≤1$, we show that there exists a global smooth solution $(ρ, u)$ when $curl u_0≡ 0$, while for $λ>1$, in general, the solution $(ρ, u)$ will blow up in finite time. Therefore, $λ=1$ appears to be the critical value for the global existence of small amplitude smooth solutions.