This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables. When the semilinear terms contain at least two  waves with different propagation speeds, we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data. Furthermore, we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearties.