# 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:
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##### 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), 563-577.  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(3) (2008), 573-587.  Google Scholar [3] V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics", American Mathematical Society Colloquium Publications, 49 (Providence, RI: American Mathematical Society), 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(B) (2000), 322-334.  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), 1169-1187.  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), 63-81.  Google Scholar [7] M. Samsonov and E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic wave guides in external media, in "Nonlinear Waves in Active Media", Springer, Berlin, (1989),99-104.  Google Scholar [8] A. M. Samsonov, Nonlinear strain waves in elastic waveguide, "Nonlinear Waves in Solids, in Cism Courses and Lecture", (eds. A. Jeffery and J. Engelbrechet), vol. 341, Springer, Wien, 1994.  Google Scholar [9] A. M. Samsonov, On Some Exact Travelling Wave Solutions for Nonlinear Hyperbolic Equation, in "Pitman Research Notes in Mathematics Series", vol. 227, Longman, (1993), 123-132.  Google Scholar [10] S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. TMA., 64 (2006), 159-173.  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), 739-751.  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), 4977-4983.  Google Scholar [13] Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal. TMA., 70 (2009), 2132-2142.  Google Scholar [14] Z. J. Yang, A global attractor for the elastic waveguide model in $R^N$, Nonlinear Anal. TMA., 74 (2011), 6640-6661.  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), 351-392.  Google Scholar
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