• Previous Article
    A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system
doi: 10.3934/dcdsb.2020147

Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Wenjun Liu

Received  April 2019 Revised  November 2019 Published  May 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation:
$ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $
together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup
$ S(t) $
.
Citation: Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020147
References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

M. Al-GwaizV. Benci and F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), 18-34.  doi: 10.1016/j.na.2014.04.011.  Google Scholar

[3]

O. H. Ammann, T. von Karman and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, 1941. Google Scholar

[4]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engng. Struct. Dyn., 23 (1994), 1351-1367.  doi: 10.1002/eqe.4290231206.  Google Scholar

[5]

A. Benaissa and M. Bahlil, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411-1437.  doi: 10.11650/tjm.18.2014.3586.  Google Scholar

[6]

A. Benaissa, A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514, 19 pp. doi: 10.1063/1.4765046.  Google Scholar

[7]

A. BenaissaA. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5 (2014), 1-26.  doi: 10.1504/IJDSDE.2014.067080.  Google Scholar

[8]

E. BerchioA. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions, Nonlinear Anal. Real World Appl., 28 (2016), 91-125.  doi: 10.1016/j.nonrwa.2015.09.005.  Google Scholar

[9]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.  Google Scholar

[10]

M. M. Cavalcanti et al., Stabilization of a suspension bridge with locally distributed damping, Math. Control Signals Systems, 30 (2018), Art. 20, 39 pp. doi: 10.1007/s00498-018-0226-0.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.  doi: 10.1016/j.jde.2006.09.019.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[14]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

[15]

L. H. Fatori et al., Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 259 (2015), 4831-4862. doi: 10.1016/j.jde.2015.06.026.  Google Scholar

[16]

A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Disc. Cont. Dynam. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879.  Google Scholar

[17]

F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Differential Equations, 2013 (2013), 47 pp.  Google Scholar

[18]

F. Gazzola and Y. Wang, Modeling suspension bridges through the von Kármán quasilinear plate equations, in Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, Cham., 86 (2015), 269–297 doi: 10.1007/978-3-319-19902-3_18.  Google Scholar

[19]

Z.-J. Han and G.-Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17 (2011), 552-574.  doi: 10.1051/cocv/2010009.  Google Scholar

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[21]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[22]

W. Lacarbonara, Nonlinear Structural Mechanics, Springer, New York, 2013. doi: 10.1007/978-1-4419-1276-3.  Google Scholar

[23]

W. J. Liu and H. F. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 67, 35 pp. doi: 10.1007/s00030-017-0491-5.  Google Scholar

[24]

S. A. Messaoudi et al., The global attractor for a suspension bridge with memory and partially hinged boundary conditions, ZAMM Z. Angew. Math. Mech., 97 (2017), 159-172. doi: 10.1002/zamm.201600034.  Google Scholar

[25]

N. Mezouar, M. Abdelli and A. Rachah, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback, Electron. J. Differential Equations, 2017 (2017), Paper No. 58, 25 pp.  Google Scholar

[26]

S. A. MessaoudiS. E. Mukiawa and E. D. Cyril, Finite dimensional global attractor for a suspension bridge problem with delay, C. R. Math. Acad. Sci. Paris, 354 (2016), 808-824.  doi: 10.1016/j.crma.2016.05.014.  Google Scholar

[27]

T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402-3412.  doi: 10.1016/j.na.2010.07.023.  Google Scholar

[28]

P. J. McKenna and C. O. Tuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745.  doi: 10.1080/00029890.2001.11919805.  Google Scholar

[29]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.  doi: 10.1007/BF00251232.  Google Scholar

[30]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[31]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar

[32]

J. Y. Park and J. R. Kang, Global attractor for hyperbolic equation with nonlinear damping and linear memory, Sci. China Math., 53 (2010), 1531-1539.  doi: 10.1007/s11425-010-3110-z.  Google Scholar

[33]

J.-Y. Park and J.-R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1.  Google Scholar

[34]

S.-H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45, 12 pp. doi: 10.1007/s00033-018-0934-9.  Google Scholar

[35]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound Vib., 307 (2007), 894-905.  doi: 10.1016/j.jsv.2007.07.036.  Google Scholar

[36] R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability, ASCE Press, 2001.  doi: 10.1061/9780784405420.  Google Scholar
[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., Vol. 68, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[38]

X. Wang, L. Yang and Q. Ma, Uniform attractors for non-autonomous suspension bridge-type equations, Bound. Value Probl., 2014 (2014), 14 pp. doi: 10.1186/1687-2770-2014-75.  Google Scholar

[39]

Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, J. Math. Anal. Appl., 418 (2014), 713-733.  doi: 10.1016/j.jmaa.2014.04.015.  Google Scholar

[40]

L. Yang and C.-K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal., 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016.  Google Scholar

[41]

Z. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal., 70 (2009), 2132-2142.  doi: 10.1016/j.na.2008.02.114.  Google Scholar

[42]

Z. Yang and X. Li, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375 (2011), 579-593.  doi: 10.1016/j.jmaa.2010.09.051.  Google Scholar

[43]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.   Google Scholar

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mecanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[2]

M. Al-GwaizV. Benci and F. Gazzola, Bending and stretching energies in a rectangular plate modeling suspension bridges, Nonlinear Anal., 106 (2014), 18-34.  doi: 10.1016/j.na.2014.04.011.  Google Scholar

[3]

O. H. Ammann, T. von Karman and G. B. Woodruff, The Failure of the Tacoma Narrows Bridge, Federal Works Agency, 1941. Google Scholar

[4]

J. M. W. Brownjohn, Observations on non-linear dynamic characteristics of suspension bridges, Earthquake Engng. Struct. Dyn., 23 (1994), 1351-1367.  doi: 10.1002/eqe.4290231206.  Google Scholar

[5]

A. Benaissa and M. Bahlil, Global existence and energy decay of solutions to a nonlinear Timoshenko beam system with a delay term, Taiwanese J. Math., 18 (2014), 1411-1437.  doi: 10.11650/tjm.18.2014.3586.  Google Scholar

[6]

A. Benaissa, A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514, 19 pp. doi: 10.1063/1.4765046.  Google Scholar

[7]

A. BenaissaA. Benguessoum and S. A. Messaoudi, Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback, Int. J. Dyn. Syst. Differ. Equ., 5 (2014), 1-26.  doi: 10.1504/IJDSDE.2014.067080.  Google Scholar

[8]

E. BerchioA. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions, Nonlinear Anal. Real World Appl., 28 (2016), 91-125.  doi: 10.1016/j.nonrwa.2015.09.005.  Google Scholar

[9]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.  Google Scholar

[10]

M. M. Cavalcanti et al., Stabilization of a suspension bridge with locally distributed damping, Math. Control Signals Systems, 30 (2018), Art. 20, 39 pp. doi: 10.1007/s00498-018-0226-0.  Google Scholar

[11]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, J. Differential Equations, 233 (2007), 42-86.  doi: 10.1016/j.jde.2006.09.019.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[14]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

[15]

L. H. Fatori et al., Long-time behavior of a class of thermoelastic plates with nonlinear strain, J. Differential Equations, 259 (2015), 4831-4862. doi: 10.1016/j.jde.2015.06.026.  Google Scholar

[16]

A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges, Disc. Cont. Dynam. Syst., 35 (2015), 5879-5908.  doi: 10.3934/dcds.2015.35.5879.  Google Scholar

[17]

F. Gazzola, Nonlinearity in oscillating bridges, Electron. J. Differential Equations, 2013 (2013), 47 pp.  Google Scholar

[18]

F. Gazzola and Y. Wang, Modeling suspension bridges through the von Kármán quasilinear plate equations, in Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, Cham., 86 (2015), 269–297 doi: 10.1007/978-3-319-19902-3_18.  Google Scholar

[19]

Z.-J. Han and G.-Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17 (2011), 552-574.  doi: 10.1051/cocv/2010009.  Google Scholar

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, J. Math. Anal. Appl., 318 (2006), 92-101.  doi: 10.1016/j.jmaa.2005.05.031.  Google Scholar

[21]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[22]

W. Lacarbonara, Nonlinear Structural Mechanics, Springer, New York, 2013. doi: 10.1007/978-1-4419-1276-3.  Google Scholar

[23]

W. J. Liu and H. F. Zhuang, Global existence, asymptotic behavior and blow-up of solutions for a suspension bridge equation with nonlinear damping and source terms, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 67, 35 pp. doi: 10.1007/s00030-017-0491-5.  Google Scholar

[24]

S. A. Messaoudi et al., The global attractor for a suspension bridge with memory and partially hinged boundary conditions, ZAMM Z. Angew. Math. Mech., 97 (2017), 159-172. doi: 10.1002/zamm.201600034.  Google Scholar

[25]

N. Mezouar, M. Abdelli and A. Rachah, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with a delay term in the non-linear internal feedback, Electron. J. Differential Equations, 2017 (2017), Paper No. 58, 25 pp.  Google Scholar

[26]

S. A. MessaoudiS. E. Mukiawa and E. D. Cyril, Finite dimensional global attractor for a suspension bridge problem with delay, C. R. Math. Acad. Sci. Paris, 354 (2016), 808-824.  doi: 10.1016/j.crma.2016.05.014.  Google Scholar

[27]

T. F. Ma and V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal., 73 (2010), 3402-3412.  doi: 10.1016/j.na.2010.07.023.  Google Scholar

[28]

P. J. McKenna and C. O. Tuama, Large torsional oscillations in suspension bridges visited again: Vertical forcing creates torsional response, Amer. Math. Monthly, 108 (2001), 738-745.  doi: 10.1080/00029890.2001.11919805.  Google Scholar

[29]

P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal., 98 (1987), 167-177.  doi: 10.1007/BF00251232.  Google Scholar

[30]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[31]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.  Google Scholar

[32]

J. Y. Park and J. R. Kang, Global attractor for hyperbolic equation with nonlinear damping and linear memory, Sci. China Math., 53 (2010), 1531-1539.  doi: 10.1007/s11425-010-3110-z.  Google Scholar

[33]

J.-Y. Park and J.-R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1.  Google Scholar

[34]

S.-H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45, 12 pp. doi: 10.1007/s00033-018-0934-9.  Google Scholar

[35]

R. H. Plaut and F. M. Davis, Sudden lateral asymmetry and torsional oscillations of section models of suspension bridges, J. Sound Vib., 307 (2007), 894-905.  doi: 10.1016/j.jsv.2007.07.036.  Google Scholar

[36] R. Scott, In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability, ASCE Press, 2001.  doi: 10.1061/9780784405420.  Google Scholar
[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci., Vol. 68, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[38]

X. Wang, L. Yang and Q. Ma, Uniform attractors for non-autonomous suspension bridge-type equations, Bound. Value Probl., 2014 (2014), 14 pp. doi: 10.1186/1687-2770-2014-75.  Google Scholar

[39]

Y. Wang, Finite time blow-up and global solutions for fourth order damped wave equations, J. Math. Anal. Appl., 418 (2014), 713-733.  doi: 10.1016/j.jmaa.2014.04.015.  Google Scholar

[40]

L. Yang and C.-K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal., 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016.  Google Scholar

[41]

Z. Yang, Global attractor for a nonlinear wave equation arising in elastic waveguide model, Nonlinear Anal., 70 (2009), 2132-2142.  doi: 10.1016/j.na.2008.02.114.  Google Scholar

[42]

Z. Yang and X. Li, Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation, J. Math. Anal. Appl., 375 (2011), 579-593.  doi: 10.1016/j.jmaa.2010.09.051.  Google Scholar

[43]

Z. Yang and Y. Wang, Global attractor for the Kirchhoff type equation with a strong dissipation, J. Differential Equations, 249 (2010), 3258-3278.   Google Scholar

[1]

Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224

[2]

Ivana Bochicchio, Claudio Giorgi, Elena Vuk. On the viscoelastic coupled suspension bridge. Evolution Equations & Control Theory, 2014, 3 (3) : 373-397. doi: 10.3934/eect.2014.3.373

[3]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

[4]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[5]

Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229

[6]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115

[7]

Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations & Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024

[8]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[9]

Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199

[10]

Manuel Fernández-Martínez. A real attractor non admitting a connected feasible open set. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 723-725. doi: 10.3934/dcdss.2019046

[11]

Michel Chipot, Karen Yeressian. On the asymptotic behavior of variational inequalities set in cylinders. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4875-4890. doi: 10.3934/dcds.2013.33.4875

[12]

Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216

[13]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635

[14]

Yuncheng You. Global attractor of the Gray-Scott equations. Communications on Pure & Applied Analysis, 2008, 7 (4) : 947-970. doi: 10.3934/cpaa.2008.7.947

[15]

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

[16]

Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781

[17]

Rana D. Parshad, Juan B. Gutierrez. On the global attractor of the Trojan Y Chromosome model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 339-359. doi: 10.3934/cpaa.2011.10.339

[18]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[19]

Igor Shevchenko, Barbara Kaltenbacher. Absorbing boundary conditions for the Westervelt equation. Conference Publications, 2015, 2015 (special) : 1000-1008. doi: 10.3934/proc.2015.1000

[20]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (45)
  • HTML views (120)
  • Cited by (0)

Other articles
by authors

[Back to Top]