# American Institute of Mathematical Sciences

July  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 and Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407
##### References:
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##### References:
 [1] Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205. [2] Kunio Hidano, An elementary proof of global or almost global existence for quasi-linear wave equations, Tohoku Math. J., 56 (2004), 271-287. [3] Fritz John and Sergiu Klainerman, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455. doi: 10.1002/cpa.3160370403. [4] Paul Kessenich, Global existence with small initial data for three-dimensional incompressible isotropic viscoelastic materials, eprint, arXiv:0903.2824v1. [5] Sergiu Klainerman, On "almost global'' solutions to quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 36 (1983), 325-344. doi: 10.1002/cpa.3160360304. [6] Sergiu Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., 23, 293-326. [7] Sergiu Klainerman and Thomas C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D, Comm. Pure Appl. Math., 49 (1996), 307-321. 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, Math. Ann., 296 (1993), 215-234. doi: 10.1007/BF01445103. [9] Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X. [10] Thomas C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math., 151 (2000), 849-874. doi: 10.2307/121050. [11] Thomas C. Sideris and Becca Thomases, Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems, J. Hyperbolic Differ. Equ., 3 (2006), 673-690. doi: 10.1142/S0219891606000975.
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