Almost global existence of nonlinear wave equations without compact support in two dimensions.

In this paper, we consider 2‐D nonlinear wave equations with small initial data of noncompact support when the quadratic terms of the nonlinearity satisfy the null conditions but the cubic ones do not. Due to the blow‐up result for the corresponding problem with the initial data of compact support, which was first obtained by S. Alinhac (The null condition for quasilinear wave equations in two space dimensions II. Amer. J. Math, 123 [2001], no. 6, 1071‐1101), this means that it is impossible to prove the global solvability in time in general. Therefore, in this case, what we can expect is to obtain the same lower bound of the lifespan as in Alinhac's work. That is, we also show the almost global existence, and the lower bound of the lifespan is ec/ε2 for sufficiently small ε>0 and some constant c>0. To achieve this goal, the methods we adopt are the "ghost weight" technique and the modified weighted L∞‐ L∞ estimates of the solutions to the linear wave equations. In addition, an alternative proof of the global existence is given for a similar case in three dimensions, using only the L∞‐ L2 estimates of the "good derivatives." [ABSTRACT FROM AUTHOR]