Blow up of solutions to 1-d Euler equations with time-dependent damping

We study the 1-d isentropic Euler equations with time-dependent damping{∂tρ+∂x(ρu)=0,∂t(ρu)+∂x(ρu2)+∂xp(ρ)=−μ(1+t)λρu,ρ|t=0=1+ερ0(x),u|t=0=εu0(x). In a previous paper [8], we have proven that, when λ=1λ=1, μ>2μ>2, the 1-D Euler equations have global existence of small data solutions. However in this paper, we will show that, when the damping, with respect to time, decays faster or equal to 21+t, the C1C1 solution of the above system will blow up in finite time. More precisely, when λ=1λ=1, 0≤μ≤20≤μ≤2 or λ>1λ>1, μ≥0μ≥0, we will give a finite upper bound for the lifespan. Combining the results in this paper and [8], we see that, when the damping decays with time like μ(1+t)λ, the critical exponents for λ,μλ,μ to separate the global existence and finite-time blow up of small data solutions are λ=1λ=1, μ=2μ=2.