  In this paper, we are concerned with the long time behavior of the piecewise
smooth solutions to the generalized Riemann problem governed by the
compressible isothermal Euler equations in two and three dimensions.
Non-existence result is established for the fan-shaped wave structure solution,
including two shocks and one contact discontinuity and which is a perturbation
of plane waves. Therefore, unlike the one-dimensional case, the
multi-dimensional plane shocks are not stable globally. What is more, the sharp
lifespan estimate is established which is the same as the lifespan estimate for
the nonlinear wave equations in both two and three space dimensions.
