  In this paper, we study the relaxation limit of the relaxing Cauchy problem
for non-isentropic compressible Euler equations with damping in
multi-dimensions. We prove that the velocity of the relaxing equations
converges weakly to that of the relaxed equations, while other variables of the
relaxing equations converges strongly to the corresponding variables of the
relaxed equations. We show that as relaxation time approaches 0, there exists
an initial layer for the ill-prepared data, the convergence of the velocity is
strong outside the layer; while there is no initial layer for the well-prepared
data, the convergence of the velocity is strong near t=0.
