Analysis Seminar

The classical wave equation is a basic model for the propagation of waves. In even space dimensions, solutions are known to develop long-lived polynomially decaying ``tails'' inside the region where the wave has passed, in contrast with the sharp finite propagation of disturbances in odd dimensions. In this talk I will discuss how such even-dimensional tails behave in the presence of forcing and nonlinear effects.

I will first consider the simplest model, the forced wave equation on $\mathbb{R}^{1+4}$ with compactly supported forcing. Under a natural $t^{-3}$ decay assumption on the forcing, which was obtained from prior work, we obtain sharp late-time asymptotics for the solutions in a fixed compact region, and matching upper bounds of tail type for solutions arising from large initial data.

In the second part of the talk, I will explain how these ideas extend to a broad class of nonlinear wave equations with quadratic or higher-order terms, as well as linear wave equations coupled with a class of dynamical coefficients. This leads to a precise description of the dominant late-time tails for such nonlinear and linear waves, showing that the even-dimensional free, linear tail is robust well beyond the constant-coefficient, homogeneous setting.