# American Institute of Mathematical Sciences

2013, 2013(special): 797-806. doi: 10.3934/proc.2013.2013.797

## Longtime dynamics for an elastic waveguide model

 1 Department of Mathematics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China, China

Received  September 2012 Published  November 2013

The paper studies the longtime dynamics for a nonlinear wave equation arising in elastic waveguide model： $u_{tt}- \Delta u-\Delta u_{tt}+\Delta^2 u- \Delta u_t -\Delta g(u)=f(x)$. It proves that the equation possesses in trajectory phase space a global trajectory attractor $\mathcal{A}^{tr}$ and the full trajectory of the equation in $\mathcal{A}^{tr}$ is of backward regularity provided that the growth exponent of nonlinearity $g(u)$ is supercritical.
Citation: Zhijian Yang, Ke Li. Longtime dynamics for an elastic waveguide model. Conference Publications, 2013, 2013 (special) : 797-806. doi: 10.3934/proc.2013.2013.797
##### References:
 [1] G. W. Chen, Y. P. Wang and S. B. Wang, Initial boundary value problem of the generalized cubic double dispersion equation,, J. Math. Anal. Appl., 299 (2004), 563.   Google Scholar [2] G. W. Chen and H. X. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation,, Acta Mathematica Scientia, 28B (2008), 573.   Google Scholar [3] V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics",, American Mathematical Society Colloquium Publications, 49 (2002).   Google Scholar [4] Z. D. Dai and B. L. Guo, Global attractor of nonlinear strain waves in elastic wave guides,, Acta Math. Sci., 20 (2000), 322.   Google Scholar [5] Y. C. Liu and R.Z. Xu, Potential well method for Cauchy problem of generalized double dispersion equations,, J. Math. Anal. Appl., 338 (2008), 1169.   Google Scholar [6] Y. C. Liu and R. Z. Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations,, Communications on Pure and Applied Analysis, 7 (2008), 63.   Google Scholar [7] M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic wave guides in external media,, in, (1989), 99.   Google Scholar [8] A. M. Samsonov, Nonlinear strain waves in elastic waveguide,, Samsonov, 341 (1994).   Google Scholar [9] A. M. Samsonov, On Some Exact Travelling Wave Solutions for Nonlinear Hyperbolic Equation,, in, 227 (1993), 123.   Google Scholar [10] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation,, Nonlinear Anal. TMA., 64 (2006), 159.   Google Scholar [11] R. Z. Xu and Y. C. Liu, Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations,, J. Math. Anal. Appl., 359 (2009), 739.   Google Scholar [12] R. Z. Xu, Y. C. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations,, Nonlinear Anal. TMA., 71 (2009), 4977.   Google Scholar [13] Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model,, Nonlinear Anal. TMA., 70 (2009), 2132.   Google Scholar [14] Z. J. Yang, A global attractor for the elastic waveguide model in $R^N$,, Nonlinear Anal. TMA., 74 (2011), 6640.   Google Scholar [15] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Disc. Cont. Dyn. System-A, 11 (2004), 351.   Google Scholar

show all references

##### References:
 [1] G. W. Chen, Y. P. Wang and S. B. Wang, Initial boundary value problem of the generalized cubic double dispersion equation,, J. Math. Anal. Appl., 299 (2004), 563.   Google Scholar [2] G. W. Chen and H. X. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation,, Acta Mathematica Scientia, 28B (2008), 573.   Google Scholar [3] V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics",, American Mathematical Society Colloquium Publications, 49 (2002).   Google Scholar [4] Z. D. Dai and B. L. Guo, Global attractor of nonlinear strain waves in elastic wave guides,, Acta Math. Sci., 20 (2000), 322.   Google Scholar [5] Y. C. Liu and R.Z. Xu, Potential well method for Cauchy problem of generalized double dispersion equations,, J. Math. Anal. Appl., 338 (2008), 1169.   Google Scholar [6] Y. C. Liu and R. Z. Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations,, Communications on Pure and Applied Analysis, 7 (2008), 63.   Google Scholar [7] M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic wave guides in external media,, in, (1989), 99.   Google Scholar [8] A. M. Samsonov, Nonlinear strain waves in elastic waveguide,, Samsonov, 341 (1994).   Google Scholar [9] A. M. Samsonov, On Some Exact Travelling Wave Solutions for Nonlinear Hyperbolic Equation,, in, 227 (1993), 123.   Google Scholar [10] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation,, Nonlinear Anal. TMA., 64 (2006), 159.   Google Scholar [11] R. Z. Xu and Y. C. Liu, Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations,, J. Math. Anal. Appl., 359 (2009), 739.   Google Scholar [12] R. Z. Xu, Y. C. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations,, Nonlinear Anal. TMA., 71 (2009), 4977.   Google Scholar [13] Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model,, Nonlinear Anal. TMA., 70 (2009), 2132.   Google Scholar [14] Z. J. Yang, A global attractor for the elastic waveguide model in $R^N$,, Nonlinear Anal. TMA., 74 (2011), 6640.   Google Scholar [15] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities,, Disc. Cont. Dyn. System-A, 11 (2004), 351.   Google Scholar
 [1] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [2] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [3] Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695 [4] Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305 [5] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155 [6] Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743 [7] Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179 [8] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 [9] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [10] Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084 [11] Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113 [12] Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094 [13] Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 [14] Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217 [15] Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227 [16] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [17] Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685 [18] Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121 [19] Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 [20] Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507

Impact Factor:

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS