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

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

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., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).

Citation: Pengyan Ding, Zhijian Yang. Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$. Communications on Pure and Applied Analysis, 2019, 18 (2) : 825-843. doi: 10.3934/cpaa.2019040
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S. S. Antman, The equation for large vibrations of strings, Amer. Math. Monthly, 87 (1980), 359-370.  doi: 10.2307/2321203.

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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.

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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.

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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.

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M. M. CavalcantiV. N. D. CavalcantiJ. 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.

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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.

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

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

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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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

P. G. PapadopoulosM. 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. 

[23]

P. G. PapadopoulosN. 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.

[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. 

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[32]

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

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[6]

M. M. CavalcantiV. N. D. CavalcantiJ. 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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

P. G. PapadopoulosM. 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. 

[23]

P. G. PapadopoulosN. 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.

[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. 

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[32]

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

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