Long-Time Existence and Behavior of Solutions to the Inhomogeneous Kinetic FPU Equation

Abstract

We study the inhomogeneous kinetic Fermi-Pasta-Ulam (FPU) equation, a nonlinear transport equation describing the evolution of phonon density distributions with four-phonon interactions. The equation combines free transport in physical space with a nonlinear collision operator acting in momentum space and exhibiting structural degeneracies. We develop a functional framework that captures the interplay between spatial transport and the degeneracies arising in the collision operator. A key ingredient of the analysis is a dispersive estimate for the transport flow, which quantifies decay effects generated by spatial propagation. Using this dispersive mechanism, we obtain improved bounds for the nonlinear collision operator and show that small solutions near the vacuum can be propagated on time scales significantly longer than those dictated by conservation laws alone. In particular, dispersion allows one to extend the classical quadratic lifespan to a quartic time scale.