NLWE with a special scale invariant damping in odd space dimension

Marcello  D'Abbicco; Sandra  Lucente

doi:10.3934/proc.2015.0312

Article Contents

2015, Volume 2015: 312-319. Doi: 10.3934/proc.2015.0312 open access

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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

Received: August 31, 2014

Revised: October 31, 2014

Published: October 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L05, 35L71.

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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.