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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 |
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., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).
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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.
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Global bifurcation results for a semilinear elliptic equation on all of $\mathbb{R}^N$, Duke Mathematical Journal, 85 (1996), 77-94.
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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. |
[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. |
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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 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. |
[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. 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.
|
[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. |
[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. 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. |
[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 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. |
[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. 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.
|
[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. |
[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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