We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation , where . Standard energy methods guarantee global existence, but do not directly say much about the behaviour of as . Note that in contrast to higher-dimensional settings, solutions to the linear equation do not exhibit decay, thus apparently ruling out perturbative methods for understanding such solutions. Nevertheless, we will show that solutions for the nonlinear equation behave differently from the linear equation, and more specifically that we have the average decay , in sharp contrast to the linear case. An unusual ingredient in our arguments is the classical Rademacher differentiation theorem that asserts that Lipschitz functions are almost everywhere differentiable.
"Asymptotic decay for a one-dimensional nonlinear wave equation." Anal. PDE 5 (2) 411 - 422, 2012. https://doi.org/10.2140/apde.2012.5.411