Analytical solutions to the compressible Euler equations with time-dependent damping and free boundaries

In this paper, we study a class of analytical solutions to the compressible Euler equations with time-dependent damping μ(1+t)λρU, which describe compressible fluids moving into outer vacuum. Under the continuous density condition across the free boundaries separating the fluid from vacuum, we construct a class of spherically symmetric and self-similar analytical solutions in R3. The global-in-time existence of such solutions is proved for μ > 0 and λ > 1. Moreover, the free boundary tends to +∞ at an algebraic rate not more than C(1 + t)2 as t → +∞.