Combined effects of two nonlinearities in lifespan of small solutions to   semi-linear wave equations
  This paper investigates the combined effects of two distinctive power-type nonlinear terms (with parameters $p,q>1$) in the lifespan of small solutions to semi-linear wave equations. We determine the full region of $(p,q)$ to admit global existence of small solutions, at least for spatial dimensions $n=2, 3$. Moreover, for many $(p,q)$ when there is no global existence, we obtain sharp lower bound of the lifespan, which is of the same order as the upper bound of the lifespan. 
Kunio Hidano, Chengbo Wang, Kazuyoshi Yokoyama
math.AP