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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 &amp; 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

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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

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[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

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