In this paper, we study the global exisitence of $L^\infty$ weak entropy solution to the initial boundary value problem for compressible Euler equations with relaxtion and the large time asymptotic behavior of the solution. Motivated by the sub-characterisitic conditions, we proposed some structural conditions on the relaxation term comparing with the pressure function. These conditions are proved to be sufficient to construct the global $L^\infty$ entropy weak solution and to prove the equilibrium state is the global attactor of all physical weak solutions. Furthermore, the convergence rate is proved to be exponential in time. The proof is based on the entropy dissipation principle.
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