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.

show all references

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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