Georgia Institute of Technology College of Sciences

In this talk, I will discuss a model four-wave kinetic equation with a power-law dispersion relation. The goal is to identify the sharp decay threshold for local well-posedness in weighted $L^\infty$ spaces.This sharp decay threshold comes from a balance between the high-frequency strength of the collision kernel and the geometry of the resonant manifold. Stronger high-frequency growth requires more decay, while stronger curvature in the dispersion relation provides more geometric gain.We prove local well-posedness above the threshold and ill-posedness below it. The main difficulty is that the gain and loss terms cannot always be treated separately. In the most delicate regimes, the sharp estimate relies on a cancellation mechanism coming from the full nonlinear structure and the resonant geometry.