## Analysis & PDE

• Anal. PDE
• Volume 5, Number 2 (2012), 411-422.

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

#### Abstract

We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $−utt+uxx=|u|p−1u$, where $p>1$. Standard energy methods guarantee global existence, but do not directly say much about the behaviour of $u(t)$ as $t→∞$. Note that in contrast to higher-dimensional settings, solutions to the linear equation $−utt+uxx=0$ 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 $L∞$ decay $limT→+∞1T ∫ 0T∥u(t)∥Lx∞(ℝ)dt=0$, 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.

#### Article information

Source
Anal. PDE, Volume 5, Number 2 (2012), 411-422.

Dates
Revised: 12 January 2011
Accepted: 7 February 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731216

Digital Object Identifier
doi:10.2140/apde.2012.5.411

Mathematical Reviews number (MathSciNet)
MR2970713

Zentralblatt MATH identifier
1273.35049

Subjects
Primary: 35L05: Wave equation

Keywords
nonlinear wave equation

#### Citation

Lindblad, Hans; Tao, Terence. Asymptotic decay for a one-dimensional nonlinear wave equation. Anal. PDE 5 (2012), no. 2, 411--422. doi:10.2140/apde.2012.5.411. https://projecteuclid.org/euclid.apde/1513731216

#### References

• J. L. Doob, Stochastic processes, Wiley, New York, 1953.
• D. Lépingle, “La variation d'ordre $p$ des semi-martingales”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36:4 (1976), 295–316.
• H. Lindblad and A. Soffer, “A remark on asymptotic completeness for the critical nonlinear Klein–Gordon equation”, Lett. Math. Phys. 73:3 (2005), 249–258.
• M. C. Reed, “Propagation of singularities for non-linear wave equations in one dimension”, Comm. Partial Differential Equations 3:2 (1978), 153–199.
• T. Tao, Structure and randomness: pages from year one of a mathematical blog, American Mathematical Society, Providence, RI, 2008.
• T. Tao, “A quantitative version of the Besicovitch projection theorem via multiscale analysis”, Proc. Lond. Math. Soc. $(3)$ 98:3 (2009), 559–584.