American Institute of Mathematical Sciences

Advanced
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

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

show all references

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

[1]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[2]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[3]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351
[4]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[5]

A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
[6]

Jean-Michel Morel, Guoshen Yu. Is SIFT scale invariant?. Inverse Problems & Imaging, 2011, 5 (1) : 115-136. doi: 10.3934/ipi.2011.5.115
[7]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[8]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[9]

Nicolas Fourrier, Irena Lasiecka. Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions. Evolution Equations & Control Theory, 2013, 2 (4) : 631-667. doi: 10.3934/eect.2013.2.631
[10]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303
[11]

Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543
[12]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[13]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[14]

Jason Metcalfe, David Spencer. Global existence for a coupled wave system related to the Strauss conjecture. Communications on Pure & Applied Analysis, 2018, 17 (2) : 593-604. doi: 10.3934/cpaa.2018032
[15]

Mengyun Liu, Chengbo Wang. Global existence for semilinear damped wave equations in relation with the Strauss conjecture. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 709-724. doi: 10.3934/dcds.2020058
[16]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022
[17]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[18]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[19]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[20]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029

 Impact Factor: 

Tools

Metrics

  • PDF downloads (132)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences