The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided the initial data are compactly supported sufficiently small Sobolev norm. In this work, obtained an upper bound polynomial growth time for top-order energy of solutions. A natural question then arises whether time-growth is true phenomenon, despite possible conservation basic energy. present paper, we establish solutions theorem remains globally bounded time.