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), 563-577.  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

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.  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

[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, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[6]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015

[7]

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

[8]

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

[9]

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

[10]

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

[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, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084

[13]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[14]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410

[15]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043

[16]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031

[17]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, 2021, 29 (5) : 3017-3030. doi: 10.3934/era.2021024

[18]

Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224

[19]

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, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939

[20]

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

 Impact Factor: 

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]