# American Institute of Mathematical Sciences

2015, 14(4): 1407-1442. doi: 10.3934/cpaa.2015.14.1407

## Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms

 1 Department of Mathematics, University of California, Santa Barbara, CA 93106, United States, United States

Received  December 2013 Revised  April 2014 Published  April 2015

The existence of global small $\mathcal O(\varepsilon )$ solutions to quadratically nonlinear wave equations in three space dimensions under the null condition is shown to be stable under the simultaneous addition of small $\mathcal O(\nu)$ viscous dissipation and $\mathcal O(\delta)$ non-null quadratic nonlinearities, provided that $\varepsilon \delta/\nu\ll 1$. When this condition is not met, small solutions exist almost globally'', and in certain parameter ranges, the addition of dissipation enhances the lifespan.
Citation: Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407
##### References:
 [1] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 267. doi: 10.1002/cpa.3160390205. [2] Kunio Hidano, An elementary proof of global or almost global existence for quasi-linear wave equations,, \emph{Tohoku Math. J.}, 56 (2004), 271. [3] Fritz John and Sergiu Klainerman, Almost global existence to nonlinear wave equations in three space dimensions,, \emph{Comm. Pure Appl. Math.}, 37 (1984), 443. doi: 10.1002/cpa.3160370403. [4] Paul Kessenich, Global existence with small initial data for three-dimensional incompressible isotropic viscoelastic materials,, eprint, (). [5] Sergiu Klainerman, On "almost global'' solutions to quasilinear wave equations in three space dimensions,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 325. doi: 10.1002/cpa.3160360304. [6] Sergiu Klainerman, The null condition and global existence to nonlinear wave equations,, in \emph{Nonlinear Systems of Partial Differential Equations in Applied Mathematics}, 23 (1984), 293. [7] Sergiu Klainerman and Thomas C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 307. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. [8] Takayuki Kobayashi, Hartmut Pecher and Yoshihiro Shibata, On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity,, \emph{Math. Ann.}, 296 (1993), 215. doi: 10.1007/BF01445103. [9] Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations,, \emph{Nonlinear Anal.}, 9 (1985), 399. doi: 10.1016/0362-546X(85)90001-X. [10] Thomas C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, \emph{Ann. of Math.}, 151 (2000), 849. doi: 10.2307/121050. [11] Thomas C. Sideris and Becca Thomases, Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems,, \emph{J. Hyperbolic Differ. Equ.}, 3 (2006), 673. doi: 10.1142/S0219891606000975.

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##### References:
 [1] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, \emph{Comm. Pure Appl. Math.}, 39 (1986), 267. doi: 10.1002/cpa.3160390205. [2] Kunio Hidano, An elementary proof of global or almost global existence for quasi-linear wave equations,, \emph{Tohoku Math. J.}, 56 (2004), 271. [3] Fritz John and Sergiu Klainerman, Almost global existence to nonlinear wave equations in three space dimensions,, \emph{Comm. Pure Appl. Math.}, 37 (1984), 443. doi: 10.1002/cpa.3160370403. [4] Paul Kessenich, Global existence with small initial data for three-dimensional incompressible isotropic viscoelastic materials,, eprint, (). [5] Sergiu Klainerman, On "almost global'' solutions to quasilinear wave equations in three space dimensions,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 325. doi: 10.1002/cpa.3160360304. [6] Sergiu Klainerman, The null condition and global existence to nonlinear wave equations,, in \emph{Nonlinear Systems of Partial Differential Equations in Applied Mathematics}, 23 (1984), 293. [7] Sergiu Klainerman and Thomas C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D,, \emph{Comm. Pure Appl. Math.}, 49 (1996), 307. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H. [8] Takayuki Kobayashi, Hartmut Pecher and Yoshihiro Shibata, On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity,, \emph{Math. Ann.}, 296 (1993), 215. doi: 10.1007/BF01445103. [9] Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations,, \emph{Nonlinear Anal.}, 9 (1985), 399. doi: 10.1016/0362-546X(85)90001-X. [10] Thomas C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, \emph{Ann. of Math.}, 151 (2000), 849. doi: 10.2307/121050. [11] Thomas C. Sideris and Becca Thomases, Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems,, \emph{J. Hyperbolic Differ. Equ.}, 3 (2006), 673. doi: 10.1142/S0219891606000975.

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