doi: 10.3934/eect.2022009
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Blow-up of solutions to a viscoelastic wave equation with nonlocal damping

Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China

* Corresponding author: Hongwei Zhang

Received  August 2021 Revised  December 2021 Early access February 2022

Fund Project: This work is supported by National Natural Science Foundation of China (No.11801145) and the Innovative Funds Plan of Henan University of Technology (2020ZKCJ09)

The viscoelastic wave equation with nonlinear nonlocal weak damping is considered. The local existence of solutions is established. Under arbitrary positive initial energy, a finite-time blow-up result is proved by a new modified concavity method.

Citation: Donghao Li, Hongwei Zhang, Shuo Liu, Qingiyng Hu. Blow-up of solutions to a viscoelastic wave equation with nonlocal damping. Evolution Equations and Control Theory, doi: 10.3934/eect.2022009
References:
[1]

M. AkilH. BadawiS. Nicaise and A. Wehbe, Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Meth. Appl. Sci., 44 (2021), 6950-6981.  doi: 10.1002/mma.7235.

[2]

A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-Up in Nonlinear Sobolev Type Equations(De Gruyter Ser. Nonlinear Anal. Appl.; V. 15), De Gruyter, Berlin, 2011. doi: 10.1515/9783110255294.

[3]

G. AutuoriF. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Equ., 57 (2012), 379-395.  doi: 10.1080/17476933.2011.592584.

[4]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.

[5]

V. BarbuI. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611.  doi: 10.1090/S0002-9947-05-03880-8.

[6]

V. BarbuI. Lasiecka and M. A. Rammaha, Blow-up of generalized solutions to wave equations with degenerate damping and source terms, Indiana Univ. Math. J., 56 (2007), 995-1022.  doi: 10.1512/iumj.2007.56.2990.

[7]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. Jorge Silva and V. Narciso, Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, J. Differential Equations, 290 (2021), 190-222.  doi: 10.1016/j.jde.2021.04.028.

[8]

M. M. CavalcantiV. N. Domingos Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510. 

[9]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and A. Y. de Souza Franco, Exponential stability for the wave model with localized memory in a past history framework, J. Differential Equations, 264 (2018), 6535-6584.  doi: 10.1016/j.jde.2018.01.044.

[10]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and C. M. Webler, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219 (2017), 189-213.  doi: 10.1007/s11856-017-1478-y.

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, Journal of Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[12]

M. A. J. da Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[13]

X. Han and M. Wang, Global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source, Math. Nachr., 284 (2011), 703-716.  doi: 10.1002/mana.200810168.

[14]

Q. Hu and H. Zhang, Blowup and asymptotic stabiity of weak solutions to wave equations with nonlinear degenerate damping and source terms., Electron. J. Differential Equations, 2007 (2007), No. 76, 10 pp.

[15]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential and Integral Equations, 27 (2014), 931-948. 

[16]

M. A. Jorge Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete and Continuous Dynamical Systems, 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985.

[17]

M. O. Korpusov, On blowup of solutions to a Kirchhoff type dissipative wave equation with a source and positive energy, Siberian Mathematical Journal, 53 (2012), 702-717.  doi: 10.1134/S003744661204012X.

[18]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential and Integral Equations, 10 (1997), 1075-1092. 

[19]

Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, Journal of Differential Equations, 268 (2020), 7741-7773.  doi: 10.1016/j.jde.2019.11.084.

[20]

Y. LiZ. Yang and F. Da, Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping, Discrete Contin. Dyn. Syst., 39 (2019), 5975-6000.  doi: 10.3934/dcds.2019261.

[21]

Y. LiZ. Yang and P. Ding, Regular solutions and strong attractors for the Kirchhoff wave model with structural nonlinear damping, Applied Mathematics Letters, 104 (2020), 106258.  doi: 10.1016/j.aml.2020.106258.

[22]

G. Liu, Global nonexistence for abstract evolution equations with dissipation, Nonlinear Anal. Real World Appl., 22 (2015), 225-235.  doi: 10.1016/j.nonrwa.2014.09.002.

[23]

G. Liu and H. Zhang, Blow up at infinity of solutions for integro-differential equation, Appl. Math. Comput., 230 (2014), 303-314.  doi: 10.1016/j.amc.2013.12.105.

[24]

S. A. Messaoudi, Blow-up of positive-initial energy solutions of a nonlinear hyperbolic equations, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.

[25]

V. Narciso, On a Kirchhoff wave model with nonlocal ninlinear damping, Evol. Equ. Control Theory, 9 (2020), 487-508.  doi: 10.3934/eect.2020021.

[26]

V. Narciso, Attractors for a plate equation with nonlocal nonlinearities, Math. Meth. Appl. Sci., 40 (2017), 3937-3954.  doi: 10.1002/mma.4275.

[27]

D. R. Pitts and M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Univ. Math. J., 51 (2002), 1479-1509.  doi: 10.1512/iumj.2002.51.2215.

[28]

P. Pucci and S. Saldi, Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractionl $p-$Laplacian operator, J. Differential Equations, 263 (2017), 2375-2418.  doi: 10.1016/j.jde.2017.02.039.

[29]

M. A. Rammaha and S. Sakuntasathien, Critically and degenerately damped systems of nonlinear wave equations with source terms, Appl. Anal., 89 (2010), 1201-1227.  doi: 10.1080/00036811.2010.483423.

[30]

M. A. Rammaha and S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal., 72 (2010), 2658-2683.  doi: 10.1016/j.na.2009.11.013.

[31]

M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc., 354 (2002), 3621-3637.  doi: 10.1090/S0002-9947-02-03034-9.

[32]

N. Takayuki, Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations, Discrete and Continuous Dynamical Systerms, 40 (2020), 2581-2591. 

[33]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171.

[34]

S.-T. Wu and C.-Y. Lin, Global nonexistence for an integro-differential equation, Math. Methods Appl. Sci., 35 (2012), 72-83.  doi: 10.1002/mma.1535.

[35]

Z. J. YangP. Ding and Z. M. Liu, Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 33 (2014), 12-17.  doi: 10.1016/j.aml.2014.02.014.

[36]

Z. J. Yang and Z. M. Liu, Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 46 (2015), 127-132.  doi: 10.1016/j.aml.2015.02.019.

[37]

H. Zhang, D. Li, W. Zhang and Q. Hu, Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source terms, Applicable Analysis, 2020. (to appear). doi: 10.1080/00036811.2020.1836354.

show all references

References:
[1]

M. AkilH. BadawiS. Nicaise and A. Wehbe, Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Meth. Appl. Sci., 44 (2021), 6950-6981.  doi: 10.1002/mma.7235.

[2]

A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-Up in Nonlinear Sobolev Type Equations(De Gruyter Ser. Nonlinear Anal. Appl.; V. 15), De Gruyter, Berlin, 2011. doi: 10.1515/9783110255294.

[3]

G. AutuoriF. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Equ., 57 (2012), 379-395.  doi: 10.1080/17476933.2011.592584.

[4]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.

[5]

V. BarbuI. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611.  doi: 10.1090/S0002-9947-05-03880-8.

[6]

V. BarbuI. Lasiecka and M. A. Rammaha, Blow-up of generalized solutions to wave equations with degenerate damping and source terms, Indiana Univ. Math. J., 56 (2007), 995-1022.  doi: 10.1512/iumj.2007.56.2990.

[7]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. Jorge Silva and V. Narciso, Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, J. Differential Equations, 290 (2021), 190-222.  doi: 10.1016/j.jde.2021.04.028.

[8]

M. M. CavalcantiV. N. Domingos Cavalcanti and T. F. Ma, Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, Differential Integral Equations, 17 (2004), 495-510. 

[9]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and A. Y. de Souza Franco, Exponential stability for the wave model with localized memory in a past history framework, J. Differential Equations, 264 (2018), 6535-6584.  doi: 10.1016/j.jde.2018.01.044.

[10]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. J. Silva and C. M. Webler, Exponential stability for the wave equation with degenerate nonlocal weak damping, Israel J. Math., 219 (2017), 189-213.  doi: 10.1007/s11856-017-1478-y.

[11]

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, Journal of Differential Equations, 252 (2012), 1229-1262.  doi: 10.1016/j.jde.2011.08.022.

[12]

M. A. J. da Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.

[13]

X. Han and M. Wang, Global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source, Math. Nachr., 284 (2011), 703-716.  doi: 10.1002/mana.200810168.

[14]

Q. Hu and H. Zhang, Blowup and asymptotic stabiity of weak solutions to wave equations with nonlinear degenerate damping and source terms., Electron. J. Differential Equations, 2007 (2007), No. 76, 10 pp.

[15]

M. A. Jorge Silva and V. Narciso, Long-time behavior for a plate equation with nonlocal weak damping, Differential and Integral Equations, 27 (2014), 931-948. 

[16]

M. A. Jorge Silva and V. Narciso, Attractors and their properties for a class of nonlocal extensible beams, Discrete and Continuous Dynamical Systems, 35 (2015), 985-1008.  doi: 10.3934/dcds.2015.35.985.

[17]

M. O. Korpusov, On blowup of solutions to a Kirchhoff type dissipative wave equation with a source and positive energy, Siberian Mathematical Journal, 53 (2012), 702-717.  doi: 10.1134/S003744661204012X.

[18]

H. Lange and G. Perla Menzala, Rates of decay of a nonlocal beam equation, Differential and Integral Equations, 10 (1997), 1075-1092. 

[19]

Y. Li and Z. Yang, Optimal attractors of the Kirchhoff wave model with structural nonlinear damping, Journal of Differential Equations, 268 (2020), 7741-7773.  doi: 10.1016/j.jde.2019.11.084.

[20]

Y. LiZ. Yang and F. Da, Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping, Discrete Contin. Dyn. Syst., 39 (2019), 5975-6000.  doi: 10.3934/dcds.2019261.

[21]

Y. LiZ. Yang and P. Ding, Regular solutions and strong attractors for the Kirchhoff wave model with structural nonlinear damping, Applied Mathematics Letters, 104 (2020), 106258.  doi: 10.1016/j.aml.2020.106258.

[22]

G. Liu, Global nonexistence for abstract evolution equations with dissipation, Nonlinear Anal. Real World Appl., 22 (2015), 225-235.  doi: 10.1016/j.nonrwa.2014.09.002.

[23]

G. Liu and H. Zhang, Blow up at infinity of solutions for integro-differential equation, Appl. Math. Comput., 230 (2014), 303-314.  doi: 10.1016/j.amc.2013.12.105.

[24]

S. A. Messaoudi, Blow-up of positive-initial energy solutions of a nonlinear hyperbolic equations, J. Math. Anal. Appl., 320 (2006), 902-915.  doi: 10.1016/j.jmaa.2005.07.022.

[25]

V. Narciso, On a Kirchhoff wave model with nonlocal ninlinear damping, Evol. Equ. Control Theory, 9 (2020), 487-508.  doi: 10.3934/eect.2020021.

[26]

V. Narciso, Attractors for a plate equation with nonlocal nonlinearities, Math. Meth. Appl. Sci., 40 (2017), 3937-3954.  doi: 10.1002/mma.4275.

[27]

D. R. Pitts and M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations, Indiana Univ. Math. J., 51 (2002), 1479-1509.  doi: 10.1512/iumj.2002.51.2215.

[28]

P. Pucci and S. Saldi, Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractionl $p-$Laplacian operator, J. Differential Equations, 263 (2017), 2375-2418.  doi: 10.1016/j.jde.2017.02.039.

[29]

M. A. Rammaha and S. Sakuntasathien, Critically and degenerately damped systems of nonlinear wave equations with source terms, Appl. Anal., 89 (2010), 1201-1227.  doi: 10.1080/00036811.2010.483423.

[30]

M. A. Rammaha and S. Sakuntasathien, Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear Anal., 72 (2010), 2658-2683.  doi: 10.1016/j.na.2009.11.013.

[31]

M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms, Trans. Amer. Math. Soc., 354 (2002), 3621-3637.  doi: 10.1090/S0002-9947-02-03034-9.

[32]

N. Takayuki, Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations, Discrete and Continuous Dynamical Systerms, 40 (2020), 2581-2591. 

[33]

E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149 (1999), 155-182.  doi: 10.1007/s002050050171.

[34]

S.-T. Wu and C.-Y. Lin, Global nonexistence for an integro-differential equation, Math. Methods Appl. Sci., 35 (2012), 72-83.  doi: 10.1002/mma.1535.

[35]

Z. J. YangP. Ding and Z. M. Liu, Global attractor for the Kirchhoff type equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 33 (2014), 12-17.  doi: 10.1016/j.aml.2014.02.014.

[36]

Z. J. Yang and Z. M. Liu, Exponential attractor for the Kirchhoff equations with strong nonlinear damping and supercritical nonlinearity, Applied Mathematics Letters, 46 (2015), 127-132.  doi: 10.1016/j.aml.2015.02.019.

[37]

H. Zhang, D. Li, W. Zhang and Q. Hu, Asymptotic stability and blow-up for the wave equation with degenerate nonlocal nonlinear damping and source terms, Applicable Analysis, 2020. (to appear). doi: 10.1080/00036811.2020.1836354.

[1]

Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072

[2]

Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093

[3]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[4]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[5]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[6]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations and Control Theory, 2022, 11 (3) : 635-648. doi: 10.3934/eect.2021019

[7]

Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130

[8]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

[9]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

[10]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[11]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917

[12]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[13]

Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106

[14]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure and Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[15]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

[16]

Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543

[17]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[18]

Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure and Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161

[19]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271

[20]

Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (270)
  • HTML views (95)
  • Cited by (0)

Other articles
by authors

[Back to Top]