# American Institute of Mathematical Sciences

November  2016, 36(11): 6557-6580. doi: 10.3934/dcds.2016084

## 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

Received  December 2015 Revised  June 2016 Published  August 2016

The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation： $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.
Citation: Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084
##### References:
 [1] J. M. Ball, Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations,, Nonlinear Science, 7 (1997), 475.  doi: 10.1007/s003329900037.  Google Scholar [2] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar [3] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbbR^3$,, Discrete Contin. Dyn. Syst., 7 (2001), 719.  doi: 10.3934/dcds.2001.7.719.  Google Scholar [4] N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains,, J. of AMS, 21 (2008), 831.  doi: 10.1090/S0894-0347-08-00596-1.  Google Scholar [5] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287.  doi: 10.2140/pjm.2002.207.287.  Google Scholar [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.  doi: 10.1017/S0004972700040296.  Google Scholar [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.  doi: 10.1016/j.jmaa.2007.04.051.  Google Scholar [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.  doi: 10.1016/j.jde.2008.02.011.  Google Scholar [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.  doi: 10.3934/dcds.2009.24.1147.  Google Scholar [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., 1354 (1988), 234.  doi: 10.1007/BFb0089601.  Google Scholar [11] S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems,, Pacific J. Math., 136 (1989), 15.  doi: 10.2140/pjm.1989.136.15.  Google Scholar [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.  doi: 10.2307/2048084.  Google Scholar [13] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, in Memories of AMS, 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar [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.   Google Scholar [15] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping,, J. Differential Equations, 252 (2012), 1229.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar [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.  Google Scholar [17] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).  doi: 10.1007/978-3-319-22903-4.  Google Scholar [18] E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent,, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051.  doi: 10.1017/S0308210500022630.  Google Scholar [19] P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions,, Nonlinearity, 29 (2016).  doi: 10.1088/0951-7715/29/4/1171.  Google Scholar [20] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equations, 247 (2009), 1120.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar [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., (2016), 00023.   Google Scholar [22] L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation,, Comm. Partial Differential Equations, 20 (1995), 1303.  doi: 10.1080/03605309508821133.  Google Scholar [23] V. Pata and M. Squassina, On the strongly damped wave equation,, Comm. Math. Phys., 253 (2005), 511.  doi: 10.1007/s00220-004-1233-1.  Google Scholar [24] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar [25] V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.   Google Scholar [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.   Google Scholar [27] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations,, Mathemaica Bohemica, 139 (2014), 657.   Google Scholar [28] A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping,, Asymptot. Anal., 87 (2014), 191.   Google Scholar [29] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations,, Doctoral dissertation, (2015).   Google Scholar [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.  doi: 10.1080/03605300008821581.  Google Scholar [31] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali diMatematica Pura ed Applicata, 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar [32] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [33] Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion,, Nonlinear Analysis, 115 (2015), 103.  doi: 10.1016/j.na.2014.12.006.  Google Scholar

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.  doi: 10.1007/s003329900037.  Google Scholar [2] J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar [3] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\mathbbR^3$,, Discrete Contin. Dyn. Syst., 7 (2001), 719.  doi: 10.3934/dcds.2001.7.719.  Google Scholar [4] N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical waves in $3D$ domains,, J. of AMS, 21 (2008), 831.  doi: 10.1090/S0894-0347-08-00596-1.  Google Scholar [5] A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities,, Pacific J. Math., 207 (2002), 287.  doi: 10.2140/pjm.2002.207.287.  Google Scholar [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.  doi: 10.1017/S0004972700040296.  Google Scholar [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.  doi: 10.1016/j.jmaa.2007.04.051.  Google Scholar [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.  doi: 10.1016/j.jde.2008.02.011.  Google Scholar [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.  doi: 10.3934/dcds.2009.24.1147.  Google Scholar [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., 1354 (1988), 234.  doi: 10.1007/BFb0089601.  Google Scholar [11] S. Chen and R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems,, Pacific J. Math., 136 (1989), 15.  doi: 10.2140/pjm.1989.136.15.  Google Scholar [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.  doi: 10.2307/2048084.  Google Scholar [13] I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, in Memories of AMS, 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar [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.   Google Scholar [15] I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping,, J. Differential Equations, 252 (2012), 1229.  doi: 10.1016/j.jde.2011.08.022.  Google Scholar [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.  Google Scholar [17] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems,, Springer, (2015).  doi: 10.1007/978-3-319-22903-4.  Google Scholar [18] E. Feireisl, Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent,, Roy. Soc. Edinburgh Sect.- A, 125 (1995), 1051.  doi: 10.1017/S0308210500022630.  Google Scholar [19] P. J. Graber and J. L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions,, Nonlinearity, 29 (2016).  doi: 10.1088/0951-7715/29/4/1171.  Google Scholar [20] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation,, J. Differential Equations, 247 (2009), 1120.  doi: 10.1016/j.jde.2009.04.010.  Google Scholar [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., (2016), 00023.   Google Scholar [22] L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation,, Comm. Partial Differential Equations, 20 (1995), 1303.  doi: 10.1080/03605309508821133.  Google Scholar [23] V. Pata and M. Squassina, On the strongly damped wave equation,, Comm. Math. Phys., 253 (2005), 511.  doi: 10.1007/s00220-004-1233-1.  Google Scholar [24] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations,, Nonlinearity, 19 (2006), 1495.  doi: 10.1088/0951-7715/19/7/001.  Google Scholar [25] V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.   Google Scholar [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.   Google Scholar [27] A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations,, Mathemaica Bohemica, 139 (2014), 657.   Google Scholar [28] A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping,, Asymptot. Anal., 87 (2014), 191.   Google Scholar [29] A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations,, Doctoral dissertation, (2015).   Google Scholar [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.  doi: 10.1080/03605300008821581.  Google Scholar [31] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali diMatematica Pura ed Applicata, 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar [32] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [33] Z. J. Yang, N. Feng and T. F. Ma, Global attracts of the generalized double dispersion,, Nonlinear Analysis, 115 (2015), 103.  doi: 10.1016/j.na.2014.12.006.  Google Scholar
 [1] 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 - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078 [2] Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273 [3] 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 & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 [4] George Avalos. Concerning the well-posedness of a nonlinearly coupled semilinear wave and beam--like equation. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 265-288. doi: 10.3934/dcds.1997.3.265 [5] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [6] 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 & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 [7] 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 [8] Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657 [9] Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020410 [10] Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094 [11] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [12] Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315 [13] 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 [14] Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393 [15] Hideo Takaoka. Global well-posedness for the Kadomtsev-Petviashvili II equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 483-499. doi: 10.3934/dcds.2000.6.483 [16] 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 [17] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [18] Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307 [19] Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & 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 & Continuous Dynamical Systems - A, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1

2019 Impact Factor: 1.338