= Almost global existence for wave equations
{title2=$T_\varepsilon\gtrsim\exp(c/\varepsilon)$}
Almost global existence means a lifespan which grows exponentially in the inverse size of small initial data, typically $T_\varepsilon\geq\exp(c/\varepsilon)-1$ for derivative-quadratic <semilinear wave equations> in three dimensions. <Commuted wave energy> and the <Klainerman-Sobolev inequality> lead to a logarithmic accumulation $\varepsilon\log(1+T)$ in a <bootstrap argument>. This implies existence up to every fixed inverse power $\varepsilon^{-N}$ when the data are small enough depending on $N$.
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