Journal of the Mathematical Society of Japan

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

Grozdena TODOROVA, Davut UĞURLU, and Borislav YORDANOV

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We show that the nonlinear wave equation u+ut3=0 is globally well-posed in radially symmetric Sobolev spaces Hradk(R3)× Hradk-1(R3) for all integers k>2. This partially extends the well-posedness in Hk(R3)× Hk-1(R3) for all k [1,2], established by Lions and Strauss[12]. As a consequence we obtain the global existence of C solutions with radial C0 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,ut)|t=0 in radially symmetric Sobolev spaces.

Article information

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

First available in Project Euclid: 13 May 2009

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Zentralblatt MATH identifier

Primary: 35L15: Initial value problems for second-order hyperbolic equations 35L70: Nonlinear second-order hyperbolic equations
Secondary: 37L05: General theory, nonlinear semigroups, evolution equations

wave equation nonlinear damping regularity


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.

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