# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021159
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## Global existence for equivalent nonlinear special scale invariant damped wave equations

 Dipartimento Interateneo di Fisica, Università degli Studi di Bari Aldo Moro, Via Amendola 173 70125 Bari, Italy

Received  August 2021 Revised  October 2021 Early access December 2021

In this paper we give the notion of equivalent damped wave equations. As an application we study global in time existence for the solution of special scale invariant damped wave equation with small data. To gain such results, without radial assumption, we deal with Klainerman vector fields. In particular we can treat some potential behind the forcing term.

Citation: Sandra Lucente. Global existence for equivalent nonlinear special scale invariant damped wave equations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021159
##### References:
 [1] M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods Appl. Sci., 38 (2015), 1032-1045.  doi: 10.1002/mma.3126. [2] M. D'Abbicco and S. Lucente, NLWE with a special scale invariant damping in odd space dimension, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, AIMS Conference. Suppl. 10 (2015), 312–319. doi: 10.3934/proc.2015.0312. [3] M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073.  doi: 10.1016/j.jde.2015.06.018. [4] R. Glassey, Existence in the large for $(\partial_t^2-\Delta)u=F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042. [5] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [6] M. Kato and M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions, Nonlinear Anal., 182 (2019), 209-225.  doi: 10.1016/j.na.2018.12.013. [7] S. Klainerman, Uniform decay estimates and Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math, 38 (1985), 321-332.  doi: 10.1002/cpa.3160380305. [8] T. Li and X. Yu, Lifespan of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations, 16 (1991), 909-940.  doi: 10.1080/03605309108820785. [9] W. Nunes do Nascimento, A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Math. Nachr., 290 (2017), 1779-1805.  doi: 10.1002/mana.201600069. [10] A. Palmieri, Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces, J. Math. Anal. Appl., 461 (2018), 1215-1240.  doi: 10.1016/j.jmaa.2018.01.063. [11] A. Palmieri, A global existence result for a semilinear scale-invariant wave equation in even dimension, Math. Methods Appl. Sci., 42 (2019), 2680-2706.  doi: 10.1002/mma.5542. [12] A. Palmieri, Global existence results for a semilinear wave equation with scale-invariant damping and mass in odd space dimension, New Tools for Nonlinear PDEs and Application, (2019), 305-369.  doi: 10.1007/978-3-030-10937-0_12. [13] A. Palmieri and M. Reissig, Semi–linear wave models with power non–linearity and scale–invariant time–dependent mass and dissipation Ⅱ, Math. Nachr., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144. [14] Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8 (1995), 135-144.

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##### References:
 [1] M. D'Abbicco, The threshold of effective damping for semilinear wave equations, Math. Methods Appl. Sci., 38 (2015), 1032-1045.  doi: 10.1002/mma.3126. [2] M. D'Abbicco and S. Lucente, NLWE with a special scale invariant damping in odd space dimension, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, AIMS Conference. Suppl. 10 (2015), 312–319. doi: 10.3934/proc.2015.0312. [3] M. D'Abbicco, S. Lucente and M. Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping, J. Differential Equations, 259 (2015), 5040-5073.  doi: 10.1016/j.jde.2015.06.018. [4] R. Glassey, Existence in the large for $(\partial_t^2-\Delta)u=F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042. [5] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [6] M. Kato and M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions, Nonlinear Anal., 182 (2019), 209-225.  doi: 10.1016/j.na.2018.12.013. [7] S. Klainerman, Uniform decay estimates and Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math, 38 (1985), 321-332.  doi: 10.1002/cpa.3160380305. [8] T. Li and X. Yu, Lifespan of classical solutions to fully nonlinear wave equations, Comm. Partial Differential Equations, 16 (1991), 909-940.  doi: 10.1080/03605309108820785. [9] W. Nunes do Nascimento, A. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Math. Nachr., 290 (2017), 1779-1805.  doi: 10.1002/mana.201600069. [10] A. Palmieri, Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces, J. Math. Anal. Appl., 461 (2018), 1215-1240.  doi: 10.1016/j.jmaa.2018.01.063. [11] A. Palmieri, A global existence result for a semilinear scale-invariant wave equation in even dimension, Math. Methods Appl. Sci., 42 (2019), 2680-2706.  doi: 10.1002/mma.5542. [12] A. Palmieri, Global existence results for a semilinear wave equation with scale-invariant damping and mass in odd space dimension, New Tools for Nonlinear PDEs and Application, (2019), 305-369.  doi: 10.1007/978-3-030-10937-0_12. [13] A. Palmieri and M. Reissig, Semi–linear wave models with power non–linearity and scale–invariant time–dependent mass and dissipation Ⅱ, Math. Nachr., 291 (2018), 1859-1892.  doi: 10.1002/mana.201700144. [14] Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Partial Differential Equations, 8 (1995), 135-144.
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