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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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). 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. 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., (2016), 00023.

[22]

L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation,, Comm. Partial Differential Equations, 20 (1995), 1303. doi: 10.1080/03605309508821133.

[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.

[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.

[25]

V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.

[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.

[27]

A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations,, Mathemaica Bohemica, 139 (2014), 657.

[28]

A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping,, Asymptot. Anal., 87 (2014), 191.

[29]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations,, Doctoral dissertation, (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. 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. doi: 10.1007/BF01762360.

[32]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (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. 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. doi: 10.1007/s003329900037.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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). 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. 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., (2016), 00023.

[22]

L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation,, Comm. Partial Differential Equations, 20 (1995), 1303. doi: 10.1080/03605309508821133.

[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.

[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.

[25]

V. Pata and S. Zelik, A remark on the damped wave equation,, Commun. Pure Appl. Anal., 5 (2006), 611.

[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.

[27]

A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations,, Mathemaica Bohemica, 139 (2014), 657.

[28]

A. Savostianov and S. Zelik, Smooth attractors for the quintic wave equations with fractional damping,, Asymptot. Anal., 87 (2014), 191.

[29]

A. Savostianov, Strichartz Estimates and Smooth Attractors of Dissipative Hyperbolic Equations,, Doctoral dissertation, (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. 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. doi: 10.1007/BF01762360.

[32]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (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. doi: 10.1016/j.na.2014.12.006.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Dalibor Pražák. Exponential attractor for the delayed logistic equation with a nonlinear diffusion. Conference Publications, 2003, 2003 (Special) : 717-726. doi: 10.3934/proc.2003.2003.717

[18]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[19]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695

[20]

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

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]