GENERAL DECAY FOR A QUASILINEAR SYSTEM OF VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING

doi:10.1016/S0252-9602(12)60101-5

Abstract

Abstract:

In this paper, we consider a system of coupled quasilinear viscoelastic equa-tions with nonlinear damping. We use the perturbed energy method to show the general decay rate estimates of energy of solutions, which extends some existing results concerning a general decay for a single equation to the case of system, and a nonlinear system of viscoelastic wave equations to a quasilinear system.

Key words: general decay, coupled quasilinear equations, viscoelastic equations, per-turbed energy method

CLC Number:

35L15

Jong Yeoul Park, Sun Hye Park. GENERAL DECAY FOR A QUASILINEAR SYSTEM OF VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING[J].Acta mathematica scientia,Series B, 2012, 32(4): 1321-1332.

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References

[1] Agre K, Rammaha M A. Systems of nonlinear wave equations with damping and source terms. Differ Integral Equ, 2006, 19: 1235–1270

[2] Alabau-Boussouira F. Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl Math Opt, 2005, 51: 61–105

[3] Andrade D, Cavalcanti M M, Cavalcanti V N D, Oquendo H P. Existence and asymptotic stability for viscoelastic evolution problems on compact manifolds. II. J Comput Anal Appl, 2006, 8: 287–301

[4] Andrade D, Mognon A. Global solutions for a system of Klein-Gordon equations with memory. Bol Soc Parana Mat Ser, 2003, 321: 127–136

[5] Arnold V I. Mathematical Methods of Classical Mechanics. Springer: New York, 1989

[6] Cavalcanti M M, Cavalcanti V N D, Ferreira J. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math Meth Appl Sci, 2001, 24: 1043–1053

[7] Cavalcanti M M, Cavalcanti V N D, Lasiecka I. Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction. J Differ Equ, 2007, 236: 407–459

[8] Cavalcanti M M, Cavalcanti V N D, Martinez P. General decay rate estimates for viscoelastic dissipative systems. Nonlinear Analysis, 2008, 68: 177–193

[9] Cavalcanti M M, Cavalcanti V N D, Santos M L. Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary. Appl Math Comput, 2004, 150: 439–465

[10] Cavalcanti M M, Cavalcanti V N D, Soriano J A. Exponential decay for the solution of semilinear vis-coelastic wave equations with localized damping. Ele J Differ Equ, 2002, 2002(44): 1–14

[11] Cavalcanti M M, Oquendo H P. Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J Contr Opt, 2003, 42: 1310–1324

[12] Guesmia A, Messaoudi S A. General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping. Math Meth Appl Sci, 2009, 32: 2102–2122

[13] Han X, Wang M. Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Analysis, 2009, 70: 3090–3098

[14] Lasiecka I. Mathematical Control Theory of Coupled PDE's. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol 75. Philadelphia, PA: SIAM, 2002

[15] Lasiecka I, Tataru D. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ Integ Equ, 1993, 6: 507–533

[16] Liu W. Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Analysis, 2009, 71: 2257–2267

[17] Messaoudi S A. General decay of solutions of a viscoelastic equation. J Math Anal Appl, 2008, 341: 1457–1467

[18] Messaoudi S A. General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Analysis, 2008, 69: 2589–2598

[19] Messaoudi S A, Tatar N -E. Exponential and polynomial decay for a quasilinear viscoelastic equation. Nonlinear Analysis, 2008, 68: 785–793

[20] Messaoudi S A, Tatar N -E. Uniform stabilization of solutions of a nonlinear system of viscoelastic equa-tions. Appl Anal, 2008, 87: 247–263

[21] Miranda M M, Medeiros L A. On the existence of global solutions of a coupled nonlinear Klien-Gordon equations. Funkcial Ekvac, 1987, 30: 147–161

[22] Park J Y, Park S H. General decay for quasilinear viscoelastic equations with nonlinear weak damping. J Math Phy, 2009, 50(8): 083505

[23] Said-Houari B. Global nonexistence of positive intial-energy solutions of a system of nonlinear wave equa-tions with damping and source terms. Differ Integ Equ, 2010, 23: 79–92

[24] Said-Houari B. Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terems. Z Angew Math Phys, 2011, 62: 115–133

[25] Said-Houari B, Messaoudi S A, Guesmia A. Genergal decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear Differ Equ Appl, 2011, 18(6): 659–684

[26] Segal I E. The global Cauchy problem for relativistic scalar field with power interactions. Bull Soc Math France, 1963, 91: 129–135

[27] Wu Shuntang. General decay of solutions for a viscoelastic equation with nonlinear damping and source terms. Acta Math Sci, 2011, 31B: 1436–1448

[28] Vicente A. Wave equation with acoustic/memory boundary conditions. Bol Soc Parana Mat, 2009, 27: 29–39

[29] Vitillaro E. Global nonexistence theorems for a class of evolution equations with dissipation. Arch Ration Mech Anal, 1999, 149: 155–182

[30] Zhang J. On the standing wave in coupled non-linear Klein-Gordon equations. Math Meth Appl Sci, 2003, 26: 11–25