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NLWE with a special scale invariant damping in odd space dimension

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  • Let $p_0(k)$ be the critical Strauss exponent for the nonlinear wave equation $u_{t t}-\Delta u=|u|^p$ in $\mathbb{R}_t\times \mathbb{R}_x^k$. In this note we prove global existence for small data radial solutions to $v_{t t}-\Delta v+2(1+t)^{-1}v_t=|v|^p$ in $\mathbb{R}_t\times \mathbb{R}_x^n$, provided that $p>p_0(n+2)$ and $n\geq5$ is odd. This result is a counterpart of the non-existence result for $p\in(1,p_0(n+2)]$ in [2]. In particular we show that the scale invariant damping term $2(1+t)^{-1}u_t$ shifts by 2 the critical exponent of NLWE.
    Mathematics Subject Classification: Primary: 35L05, 35L71.


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  • [1]

    M. D'Abbicco, The Threshold of Effective Damping for Semilinear Wave Equations, Mathematical Methods in Appl. Sci., 38, (2015), 1032-1045.


    M. D'Abbicco, S. Lucente, M. Reissig, A shift in the critical exponent for semilinear wave equations with a not effective damping, Journal of Differential Equations 259 (2015), 5040-5073.


    H. Kubo, On critical decay and power for semilinear wave equations in odd space dimensions, Dept. Math, Hokkaido Univ. Preprint series # 274, ID1404 (1997), 1-24, http://eprints3.math.sci.hokudai.ac.jp/1404.

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