The maximum principle and the theory of compensated compactness are applied to establish an existence theorem for global weak solutions to the Cauchy problem of the non-strictly hyperbolic system—a system of the compressible Euler equation with a special pressure and a source. Homogeneous system of this system was first derived by Earnshaw S. in 1858 for isentropic flow and is also called the Euler equations of one-dimensional compressible fluid flow. The key is to obtain a priori-L

∞

estimate for solutions of the Cauchy problem for the related parabolic system by using the maximum principle and give some source terms satisfying the conditions (C1)–(C3) of Theorem 1.