The Inhomogeneous Kinetic FPUT Equation near Rayleigh-Jeans Equilibria

We study the nonlinear dynamics of the spatially inhomogeneous kinetic wave equation associated with the FPUT-$β$ problem. We prove global-in-time well-posedness near nonsingular Rayleigh-Jeans equilibria and their asymptotic stability in particular in weighted $L^\infty$ norms in space and momentum, for small initial perturbations orthogonal to the collision kernel at almost every spatial point. We describe the spectrum near the origin at small spatial frequencies, identifying two simple eigenvalues with fractional and diffusive orders $|ξ|^{8/5}$ and $ξ^2$, respectively. Combining this spectral analysis with energy estimates taking into account the transport effect, we establish algebraic decay for the linear semigroup despite the absence of a spectral gap for the collision operator. We then use these estimates to obtain nonlinear stability.