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} $

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^* $.

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 and 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.

[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. ContiV. 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. DingZ. 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. GhisiM. 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. MaJ. 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. StanislavovaA. 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. ContiV. 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. DingZ. 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. GhisiM. 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. MaJ. 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. StanislavovaA. 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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