Abstract: In this paper, we study the Cauchy problem for the following quasi-linear wave equation u_tt-2ku_xxtt=β(u_xⁿ)x, where k>0 and βare real numbers, and n≥2 is an integer. We prove that for any T>0, the Cauchy problem admits a unique global smooth solution u ∈C^∞((0, T); H^∞(R))∩C ([0, T]; H²(R))∩C¹([0, T]; L²(R)) under suitable assumptions on the initial data.

Key words:  quasi-linear , wave , equation;Cauchy , problem;global , smooth , solution