Longer-than-polynomial lifespan conjecture for spherical water waves

Let

$$denote the Diophantine equation governing the relevant resonances, and let$$

denote the spherical water waves equation. Lifespan conjecture. If the Diophantine equation

has only finitely many solutions, then there is some $\alpha>0$ such that for $\varepsilon$-Cauchy data of

, the lifespan goes beyond $\varepsilon^{-(1+\alpha)}$ as $\varepsilon\to0$. Finiteness of the resonances would permit a normal-form reduction whose nonresonant component is cubic; under this conditional arithmetic assumption, the source expects lifespan beyond the scale $\varepsilon^{-1}$, but does not establish the stated bound.