Almost global existence for wave equations ($T_{\epsilon }\gtrsim \exp (c/\epsilon )$)

Almost global existence means a lifespan which grows exponentially in the inverse size of small initial data, typically $T_{\epsilon }\ge \exp (c/\epsilon )-1$ for derivative-quadratic semilinear wave equations in three dimensions. Commuted wave energy and the Klainerman-Sobolev inequality lead to a logarithmic accumulation $\epsilon \log (1+T)$ in a bootstrap argument. This implies existence up to every fixed inverse power ${\epsilon }^{-N}$ when the data are small enough depending on $N$.