Sharp decay thresholds in weighted $L^{\infty }$ for wave kinetic equations with power-law dispersion

Abstract

We study four-wave kinetic equations in space dimension three with power-law dispersion $\ddot{I}‰(p)=|p{|}^a$ and collision kernels with high-frequency growth measured by ${\hat{I}}^2$. In weighted $L^{\infty }$ spaces, we identify the sharp decay threshold $$s_c=4{\hat{I}}^2+3-\frac{a}{2}.$$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.

59 pages

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