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Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation
Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $
1. | College of Science, Henan University of Technology, Zhengzhou, 450001, China |
2. | School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China |
The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x) $. It shows that when the nonlinearity $ g(u) $ is of supercritical growth $ p $, with $ \frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+} $, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as $ t>0 $; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space $ \mathbb{R}^N $ rather than approximating physical space $ \mathbb{R}^N $ by a sequence of balls $ \Omega_R $ as usual, we break through the longstanding existed restriction on this topic for $ p: 1\leqslant p\leqslant p^* $.
References:
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F. Abergel,
Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differ. Equ., 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. V. Babin and M. I. Vishik,
Attractors of partial differential evolution equations in an unbounded domain, P. Roy. Soc. Edinb. A, 116 (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[3] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discret. Contin. Dyn. S., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[4] |
V. Belleri and V. Pata,
Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discret. Contin. Dyn. S., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
[5] |
J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations, Centro, Edizioni, Scunla Normale superiore, 2004. |
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I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differ. Equ., 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
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I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[8] |
M. Conti, V. Pata and M. Squassina,
Strongly damped wave equations on $\mathbb{R}^3$ with critical nonlinearities, Commun. Appl. Anal., 9 (2005), 161-176.
|
[9] |
P. Y. Ding and Z. J. Yang,
Attractors for the strongly damped Kirchhoff wave equation on $\mathbb{R}^N$, Commun. Pure Appl. Anal., 18 (2019), 825-843.
doi: 10.3934/cpaa.2019040. |
[10] |
P. Y. Ding, Z. J. Yang and Y. N. Li,
Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45.
doi: 10.1016/j.aml.2017.07.008. |
[11] |
M. A. Efendiev and S. Zelik,
The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Commun. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[12] |
E. Feireisl,
Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, P. Roy. Soc. Edinb. A, 125 (1995), 1051-1062.
doi: 10.1017/S0308210500022630. |
[13] |
M. Ghisi, M. Gobbino and A. Haraux,
Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, T. Am. Math. Soc., 368 (2016), 2039-2079.
doi: 10.1090/tran/6520. |
[14] |
V. Kalantarov and S. Zelik,
Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differ. Equ., 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[15] |
N. I. Karachalios and N. M. Stavrakakis,
Existence of a global attractor for semilinear dissipative wave equation on ${\mathbb R}^N$, J. Differ. Equ., 157 (1999), 183-205.
doi: 10.1006/jdeq.1999.3618. |
[16] |
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, 8 (2002), 939-951.
doi: 10.3934/dcds.2002.8.939. |
[17] |
H. Ma, J. Zhang and C. Zhong,
Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping, Discret. Contin. Dyn. S., 24 (2019), 4721-4737.
doi: 10.3934/dcdsb.2019027. |
[18] |
H. Ma and C. Zhong,
Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133.
doi: 10.1016/j.aml.2017.06.002. |
[19] |
A. Mielke and G. Schneider,
Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[20] |
A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Elsevier, Amsterdam, 2008.
doi: 10.1016/S1874-5717(08)00003-0.. |
[21] |
M. Nakao and C. S. Chen,
On global attractors for a nonlinear parabolic equation of m-Laplacian type in $\mathbb{R}^N$, Funkcialaj Ekvacioj, 50 (2007), 449-468.
doi: 10.1619/fesi.50.449. |
[22] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[23] |
M. Stanislavova, A. Stefanov and B. X. Wang,
Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on $\mathbb{R}^3$, J. Differ. Equ., 219 (2005), 451-483.
doi: 10.1016/j.jde.2005.08.004. |
[24] |
R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, NewYork, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[25] |
B. X. Wang,
Attractors for reaction-diffusion equation in unbounded domains, Phys. D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[26] |
M. H. Yang and C. Y. Sun,
Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, T. Am. Math. Soc., 361 (2009), 1069-1101.
doi: 10.1090/S0002-9947-08-04680-1. |
[27] |
Z. J. Yang,
Longtime behavior of the Kirchhoff type equation with strong damping on $ \mathbb{R}^{N}$, J. Differ. Equ., 242 (2007), 269-286.
doi: 10.1016/j.jde.2007.08.004. |
[28] |
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. |
show all references
References:
[1] |
F. Abergel,
Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differ. Equ., 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. V. Babin and M. I. Vishik,
Attractors of partial differential evolution equations in an unbounded domain, P. Roy. Soc. Edinb. A, 116 (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[3] |
J. M. Ball,
Global attractors for damped semilinear wave equations, Discret. Contin. Dyn. S., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[4] |
V. Belleri and V. Pata,
Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discret. Contin. Dyn. S., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
[5] |
J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations, Centro, Edizioni, Scunla Normale superiore, 2004. |
[6] |
I. Chueshov,
Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differ. Equ., 252 (2012), 1229-1262.
doi: 10.1016/j.jde.2011.08.022. |
[7] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[8] |
M. Conti, V. Pata and M. Squassina,
Strongly damped wave equations on $\mathbb{R}^3$ with critical nonlinearities, Commun. Appl. Anal., 9 (2005), 161-176.
|
[9] |
P. Y. Ding and Z. J. Yang,
Attractors for the strongly damped Kirchhoff wave equation on $\mathbb{R}^N$, Commun. Pure Appl. Anal., 18 (2019), 825-843.
doi: 10.3934/cpaa.2019040. |
[10] |
P. Y. Ding, Z. J. Yang and Y. N. Li,
Global attractor of the Kirchhoff wave models with strong nonlinear damping, Appl. Math. Lett., 76 (2018), 40-45.
doi: 10.1016/j.aml.2017.07.008. |
[11] |
M. A. Efendiev and S. Zelik,
The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Commun. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[12] |
E. Feireisl,
Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, P. Roy. Soc. Edinb. A, 125 (1995), 1051-1062.
doi: 10.1017/S0308210500022630. |
[13] |
M. Ghisi, M. Gobbino and A. Haraux,
Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, T. Am. Math. Soc., 368 (2016), 2039-2079.
doi: 10.1090/tran/6520. |
[14] |
V. Kalantarov and S. Zelik,
Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differ. Equ., 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[15] |
N. I. Karachalios and N. M. Stavrakakis,
Existence of a global attractor for semilinear dissipative wave equation on ${\mathbb R}^N$, J. Differ. Equ., 157 (1999), 183-205.
doi: 10.1006/jdeq.1999.3618. |
[16] |
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, 8 (2002), 939-951.
doi: 10.3934/dcds.2002.8.939. |
[17] |
H. Ma, J. Zhang and C. Zhong,
Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping, Discret. Contin. Dyn. S., 24 (2019), 4721-4737.
doi: 10.3934/dcdsb.2019027. |
[18] |
H. Ma and C. Zhong,
Attractors for the Kirchhoff equations with strong nonlinear damping, Appl. Math. Lett., 74 (2017), 127-133.
doi: 10.1016/j.aml.2017.06.002. |
[19] |
A. Mielke and G. Schneider,
Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768.
doi: 10.1088/0951-7715/8/5/006. |
[20] |
A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains, Elsevier, Amsterdam, 2008.
doi: 10.1016/S1874-5717(08)00003-0.. |
[21] |
M. Nakao and C. S. Chen,
On global attractors for a nonlinear parabolic equation of m-Laplacian type in $\mathbb{R}^N$, Funkcialaj Ekvacioj, 50 (2007), 449-468.
doi: 10.1619/fesi.50.449. |
[22] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[23] |
M. Stanislavova, A. Stefanov and B. X. Wang,
Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on $\mathbb{R}^3$, J. Differ. Equ., 219 (2005), 451-483.
doi: 10.1016/j.jde.2005.08.004. |
[24] |
R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, NewYork, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[25] |
B. X. Wang,
Attractors for reaction-diffusion equation in unbounded domains, Phys. D, 128 (1999), 41-52.
doi: 10.1016/S0167-2789(98)00304-2. |
[26] |
M. H. Yang and C. Y. Sun,
Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, T. Am. Math. Soc., 361 (2009), 1069-1101.
doi: 10.1090/S0002-9947-08-04680-1. |
[27] |
Z. J. Yang,
Longtime behavior of the Kirchhoff type equation with strong damping on $ \mathbb{R}^{N}$, J. Differ. Equ., 242 (2007), 269-286.
doi: 10.1016/j.jde.2007.08.004. |
[28] |
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. |
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