Blow-up for compressible Euler system with space-dependent damping in 1-D

Abstract

Abstract This article considers the Cauchy problem for compressible Euler system in R {\bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L 1 ( R ) {L}^{1}\left({\bf{R}}) ), then the damping is non-effective to the long-time behavior of the solution.