-
Previous Article
Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems
- DCDS Home
- This Issue
-
Next Article
Integrability of vector fields versus inverse Jacobian multipliers and normalizers
Longtime behavior of the semilinear wave equation with gentle dissipation
1. | Department of Mathematics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001 |
2. | School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China, China |
References:
[1] |
J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Science, 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[2] |
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[3] |
V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
[4] |
N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains, J. of AMS, 21 (2008), 831-845.
doi: 10.1090/S0894-0347-08-00596-1. |
[5] |
A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.
doi: 10.2140/pjm.2002.207.287. |
[6] |
A. N. Carvalho and J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463.
doi: 10.1017/S0004972700040296. |
[7] |
A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948.
doi: 10.1016/j.jmaa.2007.04.051. |
[8] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333.
doi: 10.1016/j.jde.2008.02.011. |
[9] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast dissipative nonlinearities, Discrete Continuous Dynam. Systems - A, 24 (2009), 1147-1165.
doi: 10.3934/dcds.2009.24.1147. |
[10] |
S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., Springer-Verlag, 1354 (1988), 234-256.
doi: 10.1007/BFb0089601. |
[11] |
S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.
doi: 10.2140/pjm.1989.136.15. |
[12] |
S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0 <\alpha < 1/2$, Proceedings of the American Mathematical Society, 110 (1990), 401-415.
doi: 10.2307/2048084. |
[13] |
I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, in Memories of AMS, 195, (Providence, RI: American Mathematical Society), 2008.
doi: 10.1090/memo/0912. |
[14] |
I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. |
[15] |
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. |
[16] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science and Business Media, 2010.
doi: 10.1007/978-0-387-87712-9. |
[17] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.
doi: 10.1007/978-3-319-22903-4. |
[18] |
E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051-1062.
doi: 10.1017/S0308210500022630. |
[19] |
P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions, Nonlinearity, 29 (2016), 1171.
doi: 10.1088/0951-7715/29/4/1171. |
[20] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[21] |
V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, (2016) DOI 10.1007/s00023-016-0480-y. |
[22] |
L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323.
doi: 10.1080/03605309508821133. |
[23] |
V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533.
doi: 10.1007/s00220-004-1233-1. |
[24] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.
doi: 10.1088/0951-7715/19/7/001. |
[25] |
V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616. |
[26] |
A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. |
[27] |
A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. |
[28] |
A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptot. Anal., 87 (2014), 191-221. |
[29] |
A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015. |
[30] |
H. F. Smith and C. D. Sogge, Global Strichartz estimates for non-trapping perturbations of the Laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183.
doi: 10.1080/03605300008821581. |
[31] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali diMatematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[32] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[33] |
Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106.
doi: 10.1016/j.na.2014.12.006. |
show all references
References:
[1] |
J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Science, 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[2] |
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.
doi: 10.3934/dcds.2004.10.31. |
[3] |
V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735.
doi: 10.3934/dcds.2001.7.719. |
[4] |
N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains, J. of AMS, 21 (2008), 831-845.
doi: 10.1090/S0894-0347-08-00596-1. |
[5] |
A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.
doi: 10.2140/pjm.2002.207.287. |
[6] |
A. N. Carvalho and J. W. Cholewa, Local well-posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463.
doi: 10.1017/S0004972700040296. |
[7] |
A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948.
doi: 10.1016/j.jmaa.2007.04.051. |
[8] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differential Equations, 244 (2008), 2310-2333.
doi: 10.1016/j.jde.2008.02.011. |
[9] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Damped wave equations with fast dissipative nonlinearities, Discrete Continuous Dynam. Systems - A, 24 (2009), 1147-1165.
doi: 10.3934/dcds.2009.24.1147. |
[10] |
S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems, Lecture Notes in Math., Springer-Verlag, 1354 (1988), 234-256.
doi: 10.1007/BFb0089601. |
[11] |
S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems, Pacific J. Math., 136 (1989), 15-55.
doi: 10.2140/pjm.1989.136.15. |
[12] |
S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0 <\alpha < 1/2$, Proceedings of the American Mathematical Society, 110 (1990), 401-415.
doi: 10.2307/2048084. |
[13] |
I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping, in Memories of AMS, 195, (Providence, RI: American Mathematical Society), 2008.
doi: 10.1090/memo/0912. |
[14] |
I. Chueshov, Global attractors for a class of Kirchhoff wave models with a structural nonlinear damping, J. Abstr. Differ. Equ. Appl., 1 (2010), 86-106. |
[15] |
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. |
[16] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, Springer Science and Business Media, 2010.
doi: 10.1007/978-0-387-87712-9. |
[17] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer, 2015.
doi: 10.1007/978-3-319-22903-4. |
[18] |
E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051-1062.
doi: 10.1017/S0308210500022630. |
[19] |
P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions, Nonlinearity, 29 (2016), 1171.
doi: 10.1088/0951-7715/29/4/1171. |
[20] |
V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Differential Equations, 247 (2009), 1120-1155.
doi: 10.1016/j.jde.2009.04.010. |
[21] |
V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, (2016) DOI 10.1007/s00023-016-0480-y. |
[22] |
L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323.
doi: 10.1080/03605309508821133. |
[23] |
V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533.
doi: 10.1007/s00220-004-1233-1. |
[24] |
V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.
doi: 10.1088/0951-7715/19/7/001. |
[25] |
V. Pata and S. Zelik, A remark on the damped wave equation, Commun. Pure Appl. Anal., 5 (2006), 611-616. |
[26] |
A. Savostianov, Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains, Adv. Differential Equations, 20 (2015), 495-530. |
[27] |
A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Mathemaica Bohemica, 139 (2014), 657-665. |
[28] |
A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping, Asymptot. Anal., 87 (2014), 191-221. |
[29] |
A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations, Doctoral dissertation, University of Surrey, 2015. |
[30] |
H. F. Smith and C. D. Sogge, Global Strichartz estimates for non-trapping perturbations of the Laplacian, Comm. Partial Differential Equations, 25 (2000), 2171-2183.
doi: 10.1080/03605300008821581. |
[31] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali diMatematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[32] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[33] |
Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion, Nonlinear Analysis, 115 (2015), 103-106.
doi: 10.1016/j.na.2014.12.006. |
[1] |
Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1857-1871. doi: 10.3934/cpaa.2021043 |
[2] |
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 and Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078 |
[3] |
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 |
[4] |
Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273 |
[5] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015 |
[6] |
Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 |
[7] |
George Avalos. Concerning the well-posedness of a nonlinearly coupled semilinear wave and beam--like equation. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 265-288. doi: 10.3934/dcds.1997.3.265 |
[8] |
Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 |
[9] |
Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2537-2562. doi: 10.3934/dcdsb.2021147 |
[10] |
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[11] |
Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094 |
[12] |
Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410 |
[13] |
Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657 |
[14] |
Dan-Andrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global well-posedness and scattering for Skyrme wave maps. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1923-1933. doi: 10.3934/cpaa.2012.11.1923 |
[15] |
Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2039-2064. doi: 10.3934/cpaa.2021057 |
[16] |
Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 |
[17] |
Xiaoxiao Suo, Quansen Jiu. Global well-posedness of 2D incompressible Magnetohydrodynamic equations with horizontal dissipation. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022063 |
[18] |
Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 |
[19] |
Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055 |
[20] |
Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]