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2015 | 13 | 1 |
Tytuł artykułu

Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
We consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
otrzymano
2014-12-16
zaakceptowano
2015-06-17
online
2015-06-26
Twórcy
  • Dicle University, Department of Mathematics, 21280 Diyarbakır, Turkey
Bibliografia
  • [1] S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars. Journal Applied Mechanics 1950; 17: 35-36.
  • [2] SK. Patcheu, On a global solution and asymptotic behavior for the generalized damped extensible beam equation. Journal ofDifferential Equations 1997; 135: 299-314.[WoS]
  • [3] ST. Wu, LY.Tsai,Existence and nonexistence of global solutions for a nonlinear wave equation. Taiwanese Journal of Mathematics2009; 13B(6): 2069-2091.
  • [4] Y. Zhijian, On an extensible beam equation with nonlinear damping and source terms. Journal of Differential Equations 2013; 254:3903-3927.
  • [5] JM. Ball, Stability theory for an extensible beam. Journal of Differential Equations 1973; 14:399–418.[Crossref]
  • [6] RW. Dickey, Infinite systems of nonlinear oscillation equations with linear damping. SIAM Journal on Applied Mathematics 1970;19:208–214.[Crossref]
  • [7] TF. Ma, V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms. Nonlinear Analysis2010; 73: 3402 3412.[WoS]
  • [8] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Global existence and asymptotic stability for the nonlinear andgeneralized damped extensible plate equation, Commun. Contemp. Math. 6 (2004), 705-731.[Crossref]
  • [9] V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term. Journal ofDifferential Equations 1994; 109: 295–308.
  • [10] HA. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = - Au + F (u).Trans. Amer. Math. Soc., 1974; 192: 1–21.
  • [11] HA. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM Journal onApplied Mathematics 1974; 5: 138–146.[Crossref]
  • [12] SA. Messaoudi, Blow up in a nonlinearly damped wave equation. Mathematische Nachrichten 2001; 231: 105-111.
  • [13] SA. Messaoudi, Global nonexistence in a nonlinearly damped wave equation. Applicable Analysis 2001; 80: 269–277.
  • [14] JA. Esquivel-Avila, Dynamic analysis of a nonlinear Timoshenko equation. Abstract and Applied Analysis 2011; 2010: 1-36.
  • [15] JA. Esquivel-Avila, Global attractor for a nonlinear Timoshenko equation with source terms. Mathematical Sciences 2013; 1-8.
  • [16] RA. Adams, JJF. Fournier, Sobolev Spaces. Academic Press, New York, 2003.
  • [17] M. Nakao, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term.Journal of Mathematical Analysis and Applications 1977; 58 (2): 336-343.
  • [18] K. Ono, On global solutions and blow up solutions of nonlinear Kirchhoff strings with nonlinear dissipation. Journal of MathematicalAnalysis and Applications 1997; 216: 321-342.
  • [19] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate nonlinear wave equations ofKirchhoff type with a strong dissipation. Mathematical Methods in the Applied Sciences 1997; 20: 151-177.[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_math-2015-0040
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