NLWE with a special scale invariant damping in odd space dimension

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

NLWE with a special scale invariant damping in odd space dimension

Marcello D'Abbicco ^1, and Sandra Lucente ^2,

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

Abstract
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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.

Keywords: scale invariant damping, Wave equation, Strauss exponent..

Mathematics Subject Classification: Primary: 35L05, 35L7.

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. Google Scholar
[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), 259 (2015), 5040. Google Scholar
[3]	H. Kubo, On critical decay and power for semilinear wave equations in odd space dimensions,, Dept. Math, (1997), 1. Google Scholar

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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. Google Scholar
[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), 259 (2015), 5040. Google Scholar
[3]	H. Kubo, On critical decay and power for semilinear wave equations in odd space dimensions,, Dept. Math, (1997), 1. Google Scholar

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