  In the significant work of [2], Alinhac proved the global existence of small
solutions for 2D quasilinear wave equations under the null conditions. The
proof heavily relies on the fact that the initial data have compact support
[22]. Whether this constraint can be removed or not is still unclear. In this
paper, for fully nonlinear wave equations under the null conditions, we prove
the global well-posedness for small initial data without compact support.
Moreover, we apply our result to a class of quasilinear wave equations.
