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

We study four-wave kinetic equations in space dimension three with power-law dispersion $\omega(p)=|p|^a$ and collision kernels with high-frequency growth measured by $\beta$.

In weighted $L^\infty$ spaces, we identify the sharp decay threshold $$ s_c=4\beta+3-\frac a2. $$ 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.