Asymptotic decay for a one-dimensional nonlinear wave equation

Details Asymptotic decay for a one-dimensional nonlinear wave equation Asymptotic decay for a one-dimensional nonlinear wave equation

2010-11-03

arxiv.org/abs/1011.0949v5

Mathematics

Analysis of PDEs

11 pages, no figures, to appear, Analysis & PDE. A sign error in the proof of the energy estimate in the timelike case fixed

Scientific paper

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not directly say much about the behaviour of $u(t)$ as $t \to \infty$. Note that in contrast to higher-dimensional settings, solutions to the linear equation $-u_{tt} + u_{xx} = 0$ 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 $L^\infty$ decay $\lim_{T \to +\infty} \frac{1}{T} \int_0^T \|u(t)\|_{L^\infty_x(\R)}\ dt = 0$, in sharp contrast to the linear case. An unusual ingredient in our arguments is the classical Radamacher differentiation theorem that asserts that Lipschitz functions are almost everywhere differentiable.

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