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Applicable Analysis
An International Journal
Volume 13, 1982 - Issue 3
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Original Articles

The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability

Eliana Henriques De Brito
&
Jack HaleView further author information
Pages 219-233 | Received 23 Jul 1981, Published online: 02 May 2007

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Irena Lasiecka & John Ong. (1999) Global solvability and uniform decays of solutions to quaslinear equation with nonlinear boundary dissipation. Communications in Partial Differential Equations 24:11-12, pages 2069-2107.
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Kouémou-Patcheu Solange & Solange Kouémou-Patcheu. (1997) Global exixtence and exponential decay estimates for a dampad quasilinear equation. Communications in Partial Differential Equations 22:11-12, pages 2007-2024.
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B.F. Esham$suffix/text()$suffix/text() & B.F. Weinacht. (1988) Hyperbolic-parabolic singular perturbations for scalar nonlinearities. Applicable Analysis 29:1-2, pages 19-44.
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Benjamin F. Esham$suffix/text()$suffix/text(). (1987) A singularly perturbed second order evolution equation with nonlocal nonlinearity. Numerical Functional Analysis and Optimization 9:9-10, pages 969-986.
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Chun Xiang Zhao & Feng Juan Meng. (2023) Attractor for the Extensible Beam Equation with Nonlocal Weak Damping on Time–Dependent Space. Acta Mathematica Sinica, English Series 40:4, pages 1115-1126.
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Yuming Qin, Hongli Wang & Bin Yang. (2024) Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term. Asymptotic Analysis 137:1-2, pages 85-95.
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Jinhong Zhao & Bin Guo. (2024) Energy decay and blow-up of solutions to a coupled non-degenerate Kirchhoff-Heat system with source term. Discrete and Continuous Dynamical Systems - S 0:0, pages 0-0.
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Ducival C. Pereira, Carlos A. Raposo, Celsa H. M. Maranhão & Adriano P. Cattai. (2021) Global Existence and Uniform Decay of Solutions for a Kirchhoff Beam Equation with Nonlinear Damping and Source Term. Differential Equations and Dynamical Systems 32:1, pages 101-114.
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Menglan Liao & Qingwei Li. (2023) Blow-Up of Solutions to the Fourth-Order Equation with Variable-Exponent Nonlinear Weak Damping. Mediterranean Journal of Mathematics 20:3.
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Ducival PEREİRA & Carlos RAPOSO. (2022) Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation. Communications in Advanced Mathematical Sciences 5:3, pages 137-149.
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Fengjuan Meng, Cuncai Liu & Chang Zhang. (2021) Long-time dynamics of a class of nonlocal extensible beams with time delay. Mathematics and Mechanics of Solids 27:2, pages 319-333.
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Matteo Fogato. (2022) Asymptotic finite-dimensional approximations for a class of extensible elastic systems. Mathematics in Engineering 4:4, pages 1-36.
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Chunxiang Zhao, Fengjuan Meng & Chengkui Zhong. (2022) The well-posedness and attractor on an extensible beam equation with nonlocal weak damping. Discrete and Continuous Dynamical Systems - B 0:0, pages 0.
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Honglv Ma, Jing Wang & Jun Xie. (2021) Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping. Advances in Mathematical Physics 2021, pages 1-11.
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Tlili Kadri. (2021) On the $$L^{\infty }$$-convergence of two conservative finite difference schemes for fourth-order nonlinear strain wave equations. Computational and Applied Mathematics 40:6.
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Honglv Ma, Jin Zhang & Chengkui Zhong. (2020) Attractors for the degenerate Kirchhoff wave model with strong damping: Existence and the fractal dimension. Journal of Mathematical Analysis and Applications 484:1, pages 123670.
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Fágner D. Araruna, Anderson L. A. Araujo & Aldo T. Lourêdo. (2020) Decay of solution for degenerate Kirchhoff equation with general nonlinearity. Mathematical Methods in the Applied Sciences 43:5, pages 2695-2708.
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Chunxiang Zhao, Shan Ma & Chengkui Zhong. (2020) Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity. Journal of Mathematical Physics 61:3.
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Yanbing Yang, Jiaheng Li & Tao Yu. (2019) Qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term. Applied Numerical Mathematics 141, pages 263-285.
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Fengjuan Meng, Jie Wu & Chunxiang Zhao. (2019) Time-dependent global attractor for extensible Berger equation. Journal of Mathematical Analysis and Applications 469:2, pages 1045-1069.
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Haroldo Rodrigues Clark & Ronald Ramos Guardia. (2018) Uniform stabilization of Kirchhoff problems without damping. Mathematical Methods in the Applied Sciences 41:17, pages 7667-7677.
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Fágner Dias Araruna, Frederico de Oliveira Matias, Milton de Lacerda Oliveira & Shirley Maria Santos e Souza. 2018. Recent Advances in PDEs: Analysis, Numerics and Control. Recent Advances in PDEs: Analysis, Numerics and Control 17 32 .
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Marcio Antonio Jorge da Silva & Vando Narciso. (2015) Attractors and their properties for a class of nonlocal extensible beams. Discrete & Continuous Dynamical Systems - A 35:3, pages 985-1008.
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Kosuke Ono. (2014) Global Existence and Decay Properties of Solutions for Coupled Degenerate Dissipative Hyperbolic Systems of Kirchhoff Type. Funkcialaj Ekvacioj 57:2, pages 319-337.
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Yaojun Ye. (2013) Global existence of solutions and energy decay for a Kirchhoff-type equation with nonlinear dissipation. Journal of Inequalities and Applications 2013:1.
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F.D. Araruna & R.R. Carvalho. (2013) Global solution and exponential decay for a nonlinear hyperbolic equation in Banach spaces. Nonlinear Analysis: Real World Applications 14:6, pages 2105-2115.
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A. Vicente & C.L. Frota. (2013) On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries. Journal of Mathematical Analysis and Applications 407:2, pages 328-338.
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Zhijian Yang. (2013) On an extensible beam equation with nonlinear damping and source terms. Journal of Differential Equations 254:9, pages 3903-3927.
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Marina Ghisi. (2013) Asymptotic Limits for Mildly Degenerate Kirchhoff Equations. SIAM Journal on Mathematical Analysis 45:3, pages 1886-1906.
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Marina Ghisi & Massimo Gobbino. (2011) Hyperbolic–parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation. Mathematische Annalen 354:3, pages 1079-1102.
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Zhijian Yang & Xiao Li. (2011) Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation. Journal of Mathematical Analysis and Applications 375:2, pages 579-593.
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Yang Zhijian & Wang Yunqing. (2010) Global attractor for the Kirchhoff type equation with a strong dissipation. Journal of Differential Equations 249:12, pages 3258-3278.
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Marina Ghisi & Massimo Gobbino. (2010) Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay. Journal of Differential Equations 248:2, pages 381-402.
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Taeko Yamazaki. (2009) Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation. Mathematical Methods in the Applied Sciences 32:15, pages 1893-1918.
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Marina Ghisi & Massimo Gobbino. (2008) Hyperbolic–parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates. Journal of Differential Equations 245:10, pages 2979-3007.
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J. Límaco, H.R. Clark & L.A. Medeiros. (2008) Vibrations of elastic string with nonhomogeneous material. Journal of Mathematical Analysis and Applications 344:2, pages 806-820.
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Zhijian Yang. (2007) Longtime behavior of the Kirchhoff type equation with strong damping on . Journal of Differential Equations 242:2, pages 269-286.
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Hiromichi Hashimoto & Taeko Yamazaki. (2007) Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type. Journal of Differential Equations 237:2, pages 491-525.
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Marina Ghisi. (2006) Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term. Journal of Differential Equations 230:1, pages 128-139.
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H.R. Clark, J. Límaco & M.R. Clark. (2005) Quasilinear equation in moving domain. Nonlinear Analysis: Theory, Methods & Applications 61:7, pages 1241-1256.
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M. M. CAVALCANTI, V. N. DOMINGOS CAVALCANTI & J. A. SORIANO. (2011) GLOBAL EXISTENCE AND ASYMPTOTIC STABILITY FOR THE NONLINEAR AND GENERALIZED DAMPED EXTENSIBLE PLATE EQUATION. Communications in Contemporary Mathematics 06:05, pages 705-731.
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S. PANIZZI. (2004) SPECTRAL GAP SOLUTIONS OF THE DISSIPATIVE KIRCHHOFF EQUATION. Chinese Annals of Mathematics 25:01, pages 73-86.
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Marina Ghisi. (2000) Global Solutions for Dissipative Kirchhoff Strings with m(r)=rp (p<1). Journal of Mathematical Analysis and Applications 250:1, pages 86-97.
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Solange Kouémou Patcheu. (1997) On a Global Solution and Asymptotic Behaviour for the Generalized Damped Extensible Beam Equation. Journal of Differential Equations 135:2, pages 299-314.
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Kosuke ONO. (1997) SHARP DECAY ESTIMATES OF SOLUTIONS FOR MILDLY DEGENERATE DISSIPATIVE KIRCHHOFF EQUATIONS. Kyushu Journal of Mathematics 51:2, pages 439-451.
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Kenji Nishihara. (1984) Exponential decay of solutions of some quasilinear hyperbolic equations with linear damping. Nonlinear Analysis: Theory, Methods & Applications 8:6, pages 623-636.
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Eliana Henriques de Brito. (1984) Decay estimates for the generalized damped extensible string and beam equations. Nonlinear Analysis: Theory, Methods & Applications 8:12, pages 1489-1496.
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