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December  2021, 14(12): 4213-4230. doi: 10.3934/dcdss.2021134

Nonexistence of global solutions for a class of viscoelastic wave equations

Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo 180, Col. Reynosa Tamaulipas, 02200 Azcapotzalco, CDMX, México

* Corresponding author: jaea@azc.uam.mx

Received  August 2021 Revised  October 2021 Published  December 2021 Early access  November 2021

Fund Project: The author is supported by CONACYT grant 684340 and by the Universidad Autónoma Metropolitana

We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.

Citation: Jorge A. Esquivel-Avila. Nonexistence of global solutions for a class of viscoelastic wave equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4213-4230. doi: 10.3934/dcdss.2021134
References:
[1]

J. M. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[2]

T. CazenaveY. Martel and L. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.

[3]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[4]

X. DaiC. YangS. HuangT. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.

[5]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.

[6]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970807.

[7]

L. Jie and L. Fei, Blow-up of solution for an integro-differential equation with arbitrary positive initial energy, Bound. Value Probl., 2015 (2015), Paper No. 96, 10 pp. doi: 10.1186/s13661-015-0361-1.

[8]

M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear viscoelastic problem with arbitrary positive initial energy, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 657-665. 

[9]

M. O. Korpusov, A. V. Ovchinnikov, A. G. Sveshnikov and E. V. Yushkov, Blow-Up in Nonlinear Equations of Mathematical Physics. Theory and Methods, De Gruyter Series in Nonlinear Analysis and Applications, 27, De Gruyter, Berlin, 2018. doi: 10.1515/9783110602074.

[10]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au + \mathcal{F}(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[11]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au + \mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.

[12]

G. Li, L. Hong and W. Liu, Global nonexistence of solutions for viscoelastic wave equations of Kirchhoff type with high energy, J. Funct. Spaces Appl., 2012 (2012), Paper No. 530861, 15 pp. doi: 10.1155/2012/530861.

[13]

G. Li, Y. Sun and W. Liu, On asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic Petrovsky equation with positive initial energy, J. Funct. Spaces Appl., 2013 (2013), Paper No. 905867, 7 pp. doi: 10.1155/2013/905867.

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[15]

F. Liang and H. Gao, Global existence and blow-up of solutions for a nonlinear wave equation with memory, J. Inequal. Appl., 2012 (2012), Paper No. 33, 27 pp. doi: 10.1186/1029-242X-2012-33.

[16]

M. LiaoQ. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.

[17]

Q. LinX. TianR. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.

[18]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[19]

L. Liu, F. Sun and Y. Wu, Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level, Bound. Value Probl., 2019 (2019), Paper No. 15, 18 pp. doi: 10.1186/s13661-019-1136-x.

[20]

L. Liu, F. Sun and Y. Wu, Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy, Partial Differ. Equ. Appl., 1 (2020), Paper No. 31, 18 pp. doi: 10.1007/s42985-020-00031-1.

[21]

Y. Liu and W. Li, A family of potential wells for a wave equation, Electron. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.

[22]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.

[23]

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

[24]

H. Miyazaki, Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 41 (2021), 2411-2445.  doi: 10.3934/dcds.2020370.

[25]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

[26]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[27]

F. SunL. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.

[28]

F. Tahamtani and M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Bound. Value Probl., 2012 (2012), Paper No. 50, 15 pp. doi: 10.1186/1687-2770-2012-50.

[29]

Y. Wang, A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Lett., 22 (2009), 1394-1400.  doi: 10.1016/j.aml.2009.01.052.

[30]

S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Electron. J. Differential Equations, 2006 (2006), Paper No. 45, 9 pp.

[31]

S.-T. Wu and L.-Y. Tsai, Blow-up positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwanesse J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.

[32]

R. XuY. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.

[33]

Z. Yang and G. Fan, Blow-up for the Euler-Bernoulli viscoelastic equation with a nonlinear source, Electron. J. Differential Equations, 2015 (2015), Paper No. 306, 12 pp.

[34]

Z. Yang and Z. Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differential Equations, 2016 (2016), Paper No. 332, 8 pp.

[35]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

show all references

References:
[1]

J. M. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[2]

T. CazenaveY. Martel and L. Zhao, Finite-time blowup for a Schrödinger equation with nonlinear source term, Discrete Contin. Dyn. Syst., 39 (2019), 1171-1183.  doi: 10.3934/dcds.2019050.

[3]

H. Chen and H. Xu, Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 1185-1203.  doi: 10.3934/dcds.2019051.

[4]

X. DaiC. YangS. HuangT. Yu and Y. Zhu, Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems, Electron. Res. Arch., 28 (2020), 91-102.  doi: 10.3934/era.2020006.

[5]

J. A. Esquivel-Avila, Blow-up in damped abstract nonlinear equations, Electron. Res. Arch., 28 (2020), 347-367.  doi: 10.3934/era.2020020.

[6]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics, 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970807.

[7]

L. Jie and L. Fei, Blow-up of solution for an integro-differential equation with arbitrary positive initial energy, Bound. Value Probl., 2015 (2015), Paper No. 96, 10 pp. doi: 10.1186/s13661-015-0361-1.

[8]

M. Kafini and S. A. Messaoudi, A blow-up result in a nonlinear viscoelastic problem with arbitrary positive initial energy, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 657-665. 

[9]

M. O. Korpusov, A. V. Ovchinnikov, A. G. Sveshnikov and E. V. Yushkov, Blow-Up in Nonlinear Equations of Mathematical Physics. Theory and Methods, De Gruyter Series in Nonlinear Analysis and Applications, 27, De Gruyter, Berlin, 2018. doi: 10.1515/9783110602074.

[10]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au + \mathcal{F}(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.  doi: 10.1007/BF00263041.

[11]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au + \mathcal{F}(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.

[12]

G. Li, L. Hong and W. Liu, Global nonexistence of solutions for viscoelastic wave equations of Kirchhoff type with high energy, J. Funct. Spaces Appl., 2012 (2012), Paper No. 530861, 15 pp. doi: 10.1155/2012/530861.

[13]

G. Li, Y. Sun and W. Liu, On asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic Petrovsky equation with positive initial energy, J. Funct. Spaces Appl., 2013 (2013), Paper No. 905867, 7 pp. doi: 10.1155/2013/905867.

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.

[15]

F. Liang and H. Gao, Global existence and blow-up of solutions for a nonlinear wave equation with memory, J. Inequal. Appl., 2012 (2012), Paper No. 33, 27 pp. doi: 10.1186/1029-242X-2012-33.

[16]

M. LiaoQ. Liu and H. Ye, Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations, Adv. Nonlinear Anal., 9 (2020), 1569-1591.  doi: 10.1515/anona-2020-0066.

[17]

Q. LinX. TianR. Xu and M. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2095-2107.  doi: 10.3934/dcdss.2020160.

[18]

G. Liu, The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.  doi: 10.3934/era.2020016.

[19]

L. Liu, F. Sun and Y. Wu, Blow-up of solutions for a nonlinear Petrovsky type equation with initial data at arbitrary high energy level, Bound. Value Probl., 2019 (2019), Paper No. 15, 18 pp. doi: 10.1186/s13661-019-1136-x.

[20]

L. Liu, F. Sun and Y. Wu, Finite time blow-up for a nonlinear viscoelastic Petrovsky equation with high initial energy, Partial Differ. Equ. Appl., 1 (2020), Paper No. 31, 18 pp. doi: 10.1007/s42985-020-00031-1.

[21]

Y. Liu and W. Li, A family of potential wells for a wave equation, Electron. Res. Arch., 28 (2020), 807-820.  doi: 10.3934/era.2020041.

[22]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66.  doi: 10.1002/mana.200310104.

[23]

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

[24]

H. Miyazaki, Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations, Discrete Contin. Dyn. Syst., 41 (2021), 2411-2445.  doi: 10.3934/dcds.2020370.

[25]

M. Renardy, W. J. Hrusa and J. A. Nohel, Mathematical Problems in Viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, 35. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

[26]

T. Saanouni, Global and non global solutions for a class of coupled parabolic systems, Adv. Nonlinear Anal., 9 (2020), 1383-1401.  doi: 10.1515/anona-2020-0073.

[27]

F. SunL. Liu and Y. Wu, Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level, Appl. Anal., 98 (2019), 2308-2327.  doi: 10.1080/00036811.2018.1460812.

[28]

F. Tahamtani and M. Shahrouzi, Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term, Bound. Value Probl., 2012 (2012), Paper No. 50, 15 pp. doi: 10.1186/1687-2770-2012-50.

[29]

Y. Wang, A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Lett., 22 (2009), 1394-1400.  doi: 10.1016/j.aml.2009.01.052.

[30]

S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Electron. J. Differential Equations, 2006 (2006), Paper No. 45, 9 pp.

[31]

S.-T. Wu and L.-Y. Tsai, Blow-up positive-initial-energy solutions for an integro-differential equation with nonlinear damping, Taiwanesse J. Math., 14 (2010), 2043-2058.  doi: 10.11650/twjm/1500406031.

[32]

R. XuY. Yang and Y. Liu, Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.  doi: 10.1080/00036811.2011.601456.

[33]

Z. Yang and G. Fan, Blow-up for the Euler-Bernoulli viscoelastic equation with a nonlinear source, Electron. J. Differential Equations, 2015 (2015), Paper No. 306, 12 pp.

[34]

Z. Yang and Z. Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differential Equations, 2016 (2016), Paper No. 332, 8 pp.

[35]

M. ZhangQ. ZhaoY. Liu and W. Li, Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition, Electron. Res. Arch., 28 (2020), 369-381.  doi: 10.3934/era.2020021.

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