American Institute of Mathematical Sciences

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

Longtime dynamics for an elastic waveguide model

Zhijian Yang 1, and  Ke Li 1,

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.
Keywords: longtime dynamics, global solution, Nonlinear wave equation, trajectory attractor, backward regularity..
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35G31, 35L35, 37L3.
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), 563-577.

[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.

[3]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics", American Mathematical Society Colloquium Publications, 49 (Providence, RI: American Mathematical Society), 2002.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. TMA., 64 (2006), 159-173.

[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.

[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.

[13]

Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal. TMA., 70 (2009), 2132-2142.

[14]

Z. J. Yang, A global attractor for the elastic waveguide model in $R^N$, Nonlinear Anal. TMA., 74 (2011), 6640-6661.

[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.

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), 563-577.

[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.

[3]

V. Chepyzhov and M. Vishik, "Attractors for Equations of Mathematical Physics", American Mathematical Society Colloquium Publications, 49 (Providence, RI: American Mathematical Society), 2002.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

S. B. Wang and G. W. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. TMA., 64 (2006), 159-173.

[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.

[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.

[13]

Z. J. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal. TMA., 70 (2009), 2132-2142.

[14]

Z. J. Yang, A global attractor for the elastic waveguide model in $R^N$, Nonlinear Anal. TMA., 74 (2011), 6640-6661.

[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.

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