American Institute of Mathematical Sciences

Advanced
March  2021, 20(3): 1059-1076. doi: 10.3934/cpaa.2021006

Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $

Pengyan Ding 1, and  Zhijian Yang 2,,

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

* Corresponding author

Received  May 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: The research is supported by National Natural Science Foundation of China (Grant No. 11671367) and the Doctor Foundation of Henan University of Technology, China (No. 2019BS041). The first author is supported by the Doctor Foundation of Henan University of Technology, China (Grant No. 2019BS041). The second author is supported by NSFC (Grant No. 11671367)

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^* $.

Keywords: Strongly damped wave equation, unbounded domain, well-posedness, regularity, global attractor.
Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35B33, 35B65.
Citation: Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, NewYork, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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..  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, NewYork, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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
[2]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2039-2064. doi: 10.3934/cpaa.2021057
[3]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[4]

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
[5]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[6]

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
[7]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[8]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[9]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651
[10]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[11]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227
[12]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[13]

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
[14]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657
[15]

Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923
[16]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
[17]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[18]

Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[19]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[20]

Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (172)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy