# American Institute of Mathematical Sciences

2015, 2015(special): 312-319. doi: 10.3934/proc.2015.0312

## NLWE with a special scale invariant damping in odd space dimension

 1 Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes 3900, Ribeirão Preto, SP 14040-901 2 Department of Mathematics, University of Bari, Via E. Orabona 4, Bari, BA 70125, Italy

Received  September 2014 Revised  November 2014 Published  November 2015

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.
Citation: Marcello D'Abbicco, Sandra Lucente. NLWE with a special scale invariant damping in odd space dimension. Conference Publications, 2015, 2015 (special) : 312-319. doi: 10.3934/proc.2015.0312
##### References:
 [1] M. D'Abbicco, The Threshold of Effective Damping for Semilinear Wave Equations, Mathematical Methods in Appl. Sci., 38, (2015), 1032-1045. [2] 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. [3] 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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##### References:
 [1] M. D'Abbicco, The Threshold of Effective Damping for Semilinear Wave Equations, Mathematical Methods in Appl. Sci., 38, (2015), 1032-1045. [2] 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. [3] 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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