American Institute of Mathematical Sciences

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March  2019, 18(2): 825-843. doi: 10.3934/cpaa.2019040

Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$

Pengyan Ding 1, and  Zhijian Yang 2,,

1. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2. 

School of Mathematics and Statistics, Zhengzhou University, No. 100, Science Road, Zhengzhou 450001, China

* Corresponding author

Received  March 2018 Revised  June 2018 Published  October 2018

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on $\mathbb{R}^N$: $u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$. It proves that when the growth exponent $p$ of the nonlinearity $f(u) $ is up to the supercritical range: $ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).

Keywords: Kirchhoff wave equation, weighted energy space, supercritical nonlinearity, global attractor, exponential attractor.
Mathematics Subject Classification: Primary: 37L30, 35B33, 35B40; Secondary: 35B41, 35L30.
Citation: Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040
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[22]

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Z. J. Yang, Longtime behavior of the Kirchhoff type equation with strong damping on $ \mathbb{R}^{N}$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.  Google Scholar

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Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $\mathbb{R}^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.  Google Scholar

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Z. J. Yang and X. Li, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375 (2011), 579-593.  doi: 10.1016/j.jmaa.2010.09.051.  Google Scholar

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Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.  Google Scholar

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E. Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley and Sons, Singapore, 1989.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Academic Press, New York, 2003.  Google Scholar

[2]

S. S. Antman, The equation for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.2307/2321203.  Google Scholar

[3]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh, 116A (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[5]

K. J. Brown and N. M. Stavrakakis, Global bifurcation results for a semilinear elliptic equation on all of $\mathbb{R}^N$, Duke Mathematical Journal, 85 (1996), 77-94.  doi: 10.1215/S0012-7094-96-08503-8.  Google Scholar

[6]

M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. P. Filho and J. A. Soriano, Existence and exponential decay for a Kirchhoff-Carrier model with viscosity, J. Math. Anal. Appl., 226 (1998), 40-60.  doi: 10.1006/jmaa.1998.6057.  Google Scholar

[7]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar

[9]

G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin/New York, 1976.  Google Scholar

[10]

X. Fan and S. Zhou, Kernel sections for non-autonomous strongly damped wave equations of non-degenerate Kirchhoff-type, Appl. Math. Comput., 158 (2004), 253-266.  doi: 10.1016/j.amc.2003.08.147.  Google Scholar

[11]

E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Anal., 23 (1994), 187-195.  doi: 10.1016/0362-546X(94)90041-8.  Google Scholar

[12]

N. I. Karachalios and N. M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on $\mathbb{R}^N$, J. differential Equations, 157 (1999), 183-205.  doi: 10.1006/jdeq.1999.3618.  Google Scholar

[13]

N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb{R}^N$, Discrete Continuous Dynam. Systems - A, 8 (2002), 939-951.  doi: 10.3934/dcds.2002.8.939.  Google Scholar

[14]

G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Stuttgart, 1883. Google Scholar

[15]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.  Google Scholar

[16]

G. W. Liu and S. X. Xia, Global existence and finite time blow up for a class of semilinear wave equations on $\mathbb{R}^N$, Computers and Mathematics with Applications, 70 (2015), 1345-1356.  doi: 10.1016/j.camwa.2015.07.021.  Google Scholar

[17]

T. Matsuyama and R. Ikehata, On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term, J. Math. Anal. Appl., 204 (1996), 729-753.  doi: 10.1006/jmaa.1996.0464.  Google Scholar

[18]

J. Muñoz Rivera, E. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory, Journal of Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.  Google Scholar

[19]

M. Nakao, An attractor for a nonlinear dissipative wave equation of Kirchhoff type, J. Math. Anal. Appl., 353 (2009), 652-659.  doi: 10.1016/j.jmaa.2008.09.010.  Google Scholar

[20]

K. Ono, Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137 (1997), 273-301.  doi: 10.1006/jdeq.1997.3263.  Google Scholar

[21]

K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20 (1997), 151-177.  doi: 10.1002/(SICI)1099-1476(19970125)20:2<151::AID-MMA851>3.3.CO;2-S.  Google Scholar

[22]

P. G. Papadopoulos, M. Karamolengos and A. Pappas, Global existence and energy decay for mildly degenerate Kirchhoff's equations on $\mathbb{R}^N$, Journal of Interdisciplinary Mathematics, 12 (2009), 767-783.   Google Scholar

[23]

P. G. Papadopoulos, N. L. Matiadou and S. Fatouros, Globa existence and blow-up results for an hyperbolic problem on $\mathbb{R}^N$, Applicable Analysis, 93 (2014), 475-489.  doi: 10.1080/00036811.2013.778982.  Google Scholar

[24]

P. G. Papadopoulos and N. M. Stavrakakis, Strong global attractor for a quasi-linear nonlocal wave equation on $\mathbb{R}^N$, Electronic Journal of Differential Equations, 77 (2006), 1-10.   Google Scholar

[25]

P. G. Papadopoulos and N. M. Stavrakakis, Compact invariant sets for some quasilinear nonlocal Kirchhoff strings on $\mathbb{R}^N$, Applicable Analysis, 87 (2008), 133-148.  doi: 10.1080/00036810601127418.  Google Scholar

[26]

M. Reed and B. Simon, Methods of Modern Mathematical Physics III: Scattering Theory, Academic Press, New York, 1979.  Google Scholar

[27]

B. X. Wang, Z. H. Huo, C. C. Hao and Z. H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 2011.  Google Scholar

[28]

Z. J. Yang, Longtime behavior of the Kirchhoff type equation with strong damping on $ \mathbb{R}^{N}$, J. Differential Equations, 242 (2007), 269-286.  doi: 10.1016/j.jde.2007.08.004.  Google Scholar

[29]

Z. J. Yang and P. Y. Ding, Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on $\mathbb{R}^N$, J. Math. Anal. Appl., 434 (2016), 1826-1851.  doi: 10.1016/j.jmaa.2015.10.013.  Google Scholar

[30]

Z. J. Yang and X. Li, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375 (2011), 579-593.  doi: 10.1016/j.jmaa.2010.09.051.  Google Scholar

[31]

Z. J. Yang and Y. Q. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.  doi: 10.1016/j.jde.2010.09.024.  Google Scholar

[32]

E. Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley and Sons, Singapore, 1989.  Google Scholar

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