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Global existence and uniform decay of solutions
for a system of wave equations with dispersive and dissipative terms |
| Wenjun LIU, |
| College of Mathematics
and Physics, Nanjing University of Information Science and Technology,
Nanjing 210044, China；Department of Mathematics,
Southeast University, Nanjing 210096, China; |
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Abstract In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.
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| Keywords
Global existence
uniform decay
dispersive
dissipative
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Issue Date: 05 September 2010
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Aassila M, Cavalcanti M M, Soriano J A. Asymptotic stability and energy decay rates for solutionsof the wave equation with memory in a star-shaped domain. SIAM J Control Optim, 2000, 38(5): 1581―1602
doi: 10.1137/S0363012998344981
|
|
Alabau-Boussouira F, Cannarsa P, Sforza D. Decay estimates for second order evolution equationswith memory. J Funct Anal, 2008, 254(5): 1342―1372
doi: 10.1016/j.jfa.2007.09.012
|
|
Andrade D, Mognon A. Global solutions for a systemof Klein-Gordon equations with memory. Bol Soc Paran Mate, Ser 3, 2003, 21(1/2): 127―136
|
|
Ball J. Remarkson blow up and nonexistence theorems for nonlinear evolutions equations. Quart J Math Oxford, 1977, 28: 473―486
doi: 10.1093/qmath/28.4.473
|
|
Cavalcanti M M, Domingos Cavalcanti V N, Ferreira J. Existence and uniform decay for nonlinearviscoelastic equation with strong damping. Math Meth Appl Sci, 2001, 24: 1043―1053
doi: 10.1002/mma.250
|
|
Cavalcanti M M, Domingos Cavalcanti V N, Martinez P. General decay rate estimates for viscoelasticdissipative systems. Nonlinear Anal, 2008, 68(1): 177―193
doi: 10.1016/j.na.2006.10.040
|
|
Cavalcanti M M, Domingos Cavalcanti V N, Soriano J A. Exponential decay for the solution ofsemilinear viscoelastic wave equations with localized damping. Elect J Diff Eqs, 2002, 44: 1―14
|
|
Cavalcanti M M, Oquendo H P. Frictional versus viscoelasticdamping in a semilinear wave equation. SIAM J Control Optim, 2003, 42(4): 1310―1324
doi: 10.1137/S0363012902408010
|
|
Georgiev V, Todorova G. Existence of solutions ofthe wave equation with nonlinear damping and source terms. J Diff Eqs, 1994, 109: 295―308
doi: 10.1006/jdeq.1994.1051
|
|
Lions J L. Quelques Métodes De Résolution des Problèmesaux Limites Non Linéaires. Paris: Dunod, 1969
|
|
Liu W J. The exponential stabilization of the higher-dimensional linear systemof thermoviscoelasticity. J Math Pureset Appliquees, 1998, 37: 355―386
doi: 10.1016/S0021-7824(98)80103-7
|
|
Liu W J. Uniform decay of solutions for a quasilinear system of viscoelasticequations. Nonlinear Analysis, 2009, 71: 2257―2267
doi: 10.1016/j.na.2009.01.060
|
|
Liu Y C, Li X Y. Some remarks on the equation utt ? Δu ? Δut ? Δutt = f(u). Journal of Natural Science of Heilongjiang University, 2004, 21(3): 1―6 (in Chinese)
|
|
Medeiros L A, Milla M M. Weak solutions for a systemof nonlinear Klein-Gordon equations. AnnalMatem Pura Appl, 1987, CXLVI: 173―183
|
|
Messaoudi S A. Blow up and global existence in a nonlinear viscoelastic wave equation. Mathematische Nachrichten, 2003, 260: 58―66
doi: 10.1002/mana.200310104
|
|
Messaoudi S A, Tatar N-E. Global existence and asymptoticbehavior for a nonlinear viscoelastic Problem. Math Meth Sci Research J, 2003, 7(4): 136―149
|
|
Messaoudi S A, Tatar N-E. Global existence and uniformstability of solutions for a quasilinear viscoelastic problem. Math Meth Appl Sci, 2007, 30: 665―680
doi: 10.1002/mma.804
|
|
Messaoudi S A, Tatar N-E. Uniform stabilization ofsolutions of a nonlinear system of viscoelastic equations. Applicable Analysis, 2008, 87(3): 247―263
doi: 10.1080/00036810701668394
|
|
Santos M L. Decay rates for solutions of a system of wave equations with memory. Elect J Diff Eq, 2002, 38: 1―17
|
|
Segal I E. The global Cauchy problem for relativistic scalar field with powerinteractions. Bull Soc Math France, 1963, 91: 129―135
|
|
Shang Y D. Initial boundary value problem of equation utt ?Δu ?Δut ?Δutt = f(u). Acta MathematicaeApplicatae Sinica, 2000, 23: 385―393 (in Chinese)
|
|
Sun F Q, Wang M X. Non-existence of global solutionsfor nonlinear strongly damped hyperbolic systems. Discrete Contin Dynam Systems, 2005, 12: 949―958
doi: 10.3934/dcds.2005.12.949
|
|
Sun F Q, Wang M X. Global and blow-up solutionsfor a system of nonlinear hyperbolic equations with dissipative terms. Nonlinear Anal, 2006, 64: 739―761
doi: 10.1016/j.na.2005.04.050
|
|
Vitillaro E. Globalnonexistence theorems for a class of evolution equations with dissipation. Arch Rational Mech Anal, 1999, 149: 155―182
doi: 10.1007/s002050050171
|
|
Xu R Z, Zhao X R, Shen J H. Asymptotic behaviour of solution for fourth order waveequation with dispersive and dissipative terms. Appl Math Mech (Engl Ed), 2008, 29(2): 259―262
doi: 10.1007/s10483-008-0213-y
|
|
Yu S Q. On the strongly damped wave equation with nonlinear damping and sourceterms. Electron J Qual Theory Diff Equ, 2009, 39: 1―18
|
|
Zhang H W, Chen G W. Asymptotic stability fora nonlinear evolution equation. CommentMath Univ Carolin, 2004, 45(1): 101―107
|
|
Zhang J. Onthe standing wave in coupled non-linear Klein-Gordon equations. Math Methods Appl Sci, 2003, 26: 11―25
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