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.
