## Journal of the Mathematical Society of Japan

### Regularity and scattering for the wave equation with a critical nonlinear damping

#### Abstract

We show that the nonlinear wave equation $\Box u+u_{t}^{3}=0$ is globally well-posed in radially symmetric Sobolev spaces $H^{k}_{\mathrm{rad}}(\mbi{R}^{3})\times H^{k-1}_{\mathrm{rad}}(\mbi{R}^{3})$ for all integers $k>2$. This partially extends the well-posedness in $H^{k}(\mbi{R}^{3})\times H^{k-1}(\mbi{R}^{3})$ for all $k\in [1,2]$, established by Lions and Strauss[12]. As a consequence we obtain the global existence of $C^{\infty}$ solutions with radial $C_{0}^{\infty}$ data. The regularity problem requires smoothing and non-concentration estimates in addition to standard energy estimates, since the cubic damping is critical when $k=2$. We also establish scattering results for initial data $(u,u_{t})|_{t=0}$ in radially symmetric Sobolev spaces.

#### Article information

Source
J. Math. Soc. Japan, Volume 61, Number 2 (2009), 625-649.

Dates
First available in Project Euclid: 13 May 2009

https://projecteuclid.org/euclid.jmsj/1242220725

Digital Object Identifier
doi:10.2969/jmsj/06120625

Mathematical Reviews number (MathSciNet)
MR2532904

Zentralblatt MATH identifier
1180.35363

#### Citation

TODOROVA, Grozdena; UĞURLU, Davut; YORDANOV, Borislav. Regularity and scattering for the wave equation with a critical nonlinear damping. J. Math. Soc. Japan 61 (2009), no. 2, 625--649. doi:10.2969/jmsj/06120625. https://projecteuclid.org/euclid.jmsj/1242220725

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