一类带有非线性阻尼项和源项的四阶波动方程整体解的存在性与不存在性

Abstract

Abstract:

In this paper, we consider the initial boundary value problem of the fourth order wave equation with nonlinear damping and source terms. By the combination of Galerkin approximations, Potential well and Monotonicity-Compactness methods, the global existence of solutions is obtained with the least amount of a priori estimates. Moreover, we prove that there are solutions with negative initial energy that blow up in finite time.

Key words: Wave equation, Global existence, Blow up, Damping and source terms, Monotonicity-Compactness methods

CLC Number:

O175.4

Di Huafei, Shang Yadong. Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms[J].Acta mathematica scientia,Series A, 2015, 35(3): 618-633.

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References

[1] Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch Rat Mech Anal, 1968, 30: 148-172
[2] Payne L E, Sattinger D H. Saddle points and instability on nonlinear hyperbolic equations. Israel Math J, 1975, 22: 273-303
[3] Liu Y C. On potential wells and vaccum isolating of solutions for semilinear wave equation. J Differ Equ, 2003, 1992: 155-169
[4] Liu Y C. On potential wells and applications to semilinear hyperbolic equations and parabolic equations. Nonlinear Anal, 2006, 64: 2665-2678
[5] Xu R Z. Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Quart Appl Math, 2010, 68: 459-468
[6] Georgiev V, Todorova G. Existence of a solution of the wave equation with nonlinear damping and source terms. J Differ Equ, 1994, 109: 295-308
[7] Ikehata R. Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal T M A, 1996, 27: 1165-1175
[8] Messaoudi S A. Blow up in a nonlinear damped wave equation. Math Nachr, 2005, 231: 105-111
[9] Messaoudi S A. Global nonexistence in a nonlinearly damped wave equation. Applicable Anal, 2001, 80: 269-277
[10] Messaoudi S A. Exponential dacay of solutions of a nonlinearly damped wave equation. Nonlinear Differ Equ Appl, 2005, 12: 391-399
[11] Hayes M, Saccomandi G. Finite amplitude transverse waves in special incompressible viscoelastic solids. J Elasticity, 2000, 59: 213-225
[12] Shang Y D. Initial-boundary value problem for a class of fourth order nonlinear evolution equations. Math Appl, 2000, 13(1): 7-11
[13] Shang Y D. Initial boundary value problem of equation u_tt-△u-△u_t-△u_tt=f(u). Acta Math Appl Sin, 2000, 23: 385-393
[14] Zhang H W, Hu Q Y. Existence of global weak solution and stability of a class nonlinear evolution equation. Acta Math Sci, 2004, 24A: 329-336
[15] Xie Y Q, Zhong C K. The existence of global attractors for a class nonlinear evolution equation. J Math Anal Appl, 2007, 336: 54-69
[16] Xu R Z, Zhao X R, Shen J H. Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms. Appl Math Meth, 2008, 29: 259-262
[17] Xie Y Q, Zhong C K. Asymptotic behavior of a class of nonlinear evolution equations. Nonlinear Anal, 2009, 71(11): 5095-5105
[18] Carvalho A N, Cholewa J W. Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time. Trans Amer Math Soc, 2009, 361(5): 2567-2586
[19] Sun C Y, Yang L, Duan J Q. Asymptotic behavior for a semilinear second order evolution equation. Trans Amer Math Soc, 2011, 363(11): 6085-6109
[20] Chen C S, Wang H, Zhu S L. Global attractor and decay estimates of solutions to a class of nonlinear evolution equations. Math Methods Appl Sci, 2011, 34(5): 497-508
[21] Villiaggio P. Mathematical Models for Elastic Structures. Cambridge: Cambridge University Press, 1997
[22] Xu R Z, Yang Y B. Finite time blow up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level. International J Mathematics, 2012, 23: 1250060, 10 pages
[23] Messaoudi S A. On the decay of solutions for a class of quasilinear hyperbolic equations with nonlinear damping and source terms. Math Meth Appl Sci, 2005, 28: 1819-1828
[24] Messaoudi S A. Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation. J Math Anal Appl, 2006, 320: 902-915
[25] Lasiecka I, Tataru D. Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differential Integral Equations, 1933, 6(3): 507-533
[26] Cavalcanti M M, Cavalcanti V N D, Martinez P. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J Differential Equations, 2004 203(1) 119-158
[27] Lions J L. Quelques Méthodes de Résolution des Problémes anx Limites Non Linéaires. Paris: Dunod Gauthier-Villars, 1969
[28] Xu R Z, Yang Y B, Liu Y C. Global well-posedness for strongly damped viscoelastic wave equation. Appl Anal, 2013, 92: 138-157
[29] Xu R Z, Yang Y B. Global existence and asymptotic behaviour of solutions for a class of fourth order strongly damped nonlinear wave equations. Quart Appl Math, 2013, 71: 401-415

Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms

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