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May  2022, 15(5): 1221-1232. doi: 10.3934/dcdss.2021093

On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

Department of Mathematics and Statistics, IR Center of Construction and Building Materials, KFUPM, Dhahran 31261, Saudi Arabia

* Corresponding author: Mohammad Kafini

Received  April 2021 Revised  June 2021 Published  May 2022 Early access  August 2021

Fund Project: The first author is supported by KFUPM project # SB201026

In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.

Citation: 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
References:
[1]

M. Al-Gharabli, New general decay results for a viscoelastic plate equation with a logarithmic nonlinearity, Boundary Value Problems, (2019), Paper No. 194, 21 pp. doi: 10.1186/s13661-019-01308-0.

[2]

D. Andrade, L. H. Fatori and J. E. M. Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron J. Differential Equations, (2006), No. 53, 16 pp.

[3]

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

[4]

P. Bernner and W. von Whal, Global classical solutions of nonlinear wave equations, Mathematische Zeitschrift, 176 (1981), 87-121.  doi: 10.1007/BF01258907.

[5]

M. M. CavalcantiV. N. Domingos CavalcantiT. F. Ma and J. A. Soriano, Global existence and asymptotic stability for viscoelastic problem, Differential Integral Equations, 15 (2002), 731-748. 

[6]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[7]

T. Cazenave and A. Haraux, Équations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.

[8]

V. Georgiev and G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqs., 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[9]

X. Han, Global existence of weak solution for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.

[10]

M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Applied Mathematics Letters, 21 (2008), 549-553.  doi: 10.1016/j.aml.2007.07.004.

[11]

M. Kafini and S. A. Messaoudi, A blow-up result for a viscoelastic system in $\mathbb{R}^{n}$, Elect. J. Diff. Eqs., 113 (2007), 1-7. 

[12]

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

[13]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equation, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[14]

H. A. Levine and J. Serrin, Global nonexistence theorem for quasilinear evolution equation with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.

[15]

H. A. LevineS. R. Park and J. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.  doi: 10.1006/jmaa.1998.6126.

[16]

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

[17]

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

[18]

S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231 (2001), 105-111.  doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I.

[19]

S. A. Messaoudi, Blow up in the Cauchy problem for a nonlinearly damped wave equation, Comm. On Applied. Analysis, 7 (2003), 379-386. 

[20]

S. A. Messaoudi and B. Said Houari, Blow up of solutions of a class of wave equations with nonlinear damping and source terms, Math. Methods Appl. Sci., 27 (2004), 1687-1696.  doi: 10.1002/mma.522.

[21]

S. A. Messaoudi and N.-e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nolinear Anal., 68 (2008), 785-793.  doi: 10.1016/j.na.2006.11.036.

[22]

C. X. Miao, The time space estimates and scattering at low energy for nonlinear higher order wave equations, Math. Sin. Ser. A, 38 (1995), 708-717. 

[23]

J. E. Munoz RiveraE. C. Lapa and R. Baretto, Decay rates for viscolastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[24]

H. Pecher, Die existenz reguläer Lösungen für Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Mathematische Zeitschrift, 140 (1974), 263-279.  doi: 10.1007/BF01214167.

[25]

F. Tahamatani and M. Shahrouzi, General existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source, Bound. Value Probl., 2012 (2012), 50, 15 pp. doi: 10.1186/1687-2770-2012-50.

[26]

G. Todorova, Cauchy problem for a nonlinear wave with nonlinear damping and source terms, C. R. Acad. Sci. Paris Ser. I, 326 (1998), 191-196.  doi: 10.1016/S0764-4442(97)89469-4.

[27]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.

[28]

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

[29]

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

[30]

B. Wang, Nonlinear scattering theory for a class of wave equations in $H^{s}$, J. Math. Anal. Appl., 296 (2004), 74-96.  doi: 10.1016/j.jmaa.2004.03.050.

[31]

S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Elect. J. Diff. Eqs., (2006), No. 45, 9 pp.

[32]

Y. Ye, Existence and asymptotic behavior of global solutions for a class of nonlinear higher-order wave equation, Journal of Inequalities and Applications, 2010 (2010), Article number: 394859. doi: 10.1155/2010/394859.

[33]

Y. Ye, Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term, Nonlinear Analysis, 112 (2015), 129-146.  doi: 10.1016/j.na.2014.09.001.

[34]

Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^{n}$, Applied Math Letters, 18 (2005), 281-286.  doi: 10.1016/j.aml.2003.07.018.

show all references

References:
[1]

M. Al-Gharabli, New general decay results for a viscoelastic plate equation with a logarithmic nonlinearity, Boundary Value Problems, (2019), Paper No. 194, 21 pp. doi: 10.1186/s13661-019-01308-0.

[2]

D. Andrade, L. H. Fatori and J. E. M. Rivera, Nonlinear transmission problem with a dissipative boundary condition of memory type, Electron J. Differential Equations, (2006), No. 53, 16 pp.

[3]

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

[4]

P. Bernner and W. von Whal, Global classical solutions of nonlinear wave equations, Mathematische Zeitschrift, 176 (1981), 87-121.  doi: 10.1007/BF01258907.

[5]

M. M. CavalcantiV. N. Domingos CavalcantiT. F. Ma and J. A. Soriano, Global existence and asymptotic stability for viscoelastic problem, Differential Integral Equations, 15 (2002), 731-748. 

[6]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[7]

T. Cazenave and A. Haraux, Équations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.  doi: 10.5802/afst.543.

[8]

V. Georgiev and G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqs., 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[9]

X. Han, Global existence of weak solution for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50 (2013), 275-283.  doi: 10.4134/BKMS.2013.50.1.275.

[10]

M. Kafini and S. A. Messaoudi, A blow-up result in a Cauchy viscoelastic problem, Applied Mathematics Letters, 21 (2008), 549-553.  doi: 10.1016/j.aml.2007.07.004.

[11]

M. Kafini and S. A. Messaoudi, A blow-up result for a viscoelastic system in $\mathbb{R}^{n}$, Elect. J. Diff. Eqs., 113 (2007), 1-7. 

[12]

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

[13]

H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equation, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.

[14]

H. A. Levine and J. Serrin, Global nonexistence theorem for quasilinear evolution equation with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.

[15]

H. A. LevineS. R. Park and J. Serrin, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl., 228 (1998), 181-205.  doi: 10.1006/jmaa.1998.6126.

[16]

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

[17]

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

[18]

S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231 (2001), 105-111.  doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I.

[19]

S. A. Messaoudi, Blow up in the Cauchy problem for a nonlinearly damped wave equation, Comm. On Applied. Analysis, 7 (2003), 379-386. 

[20]

S. A. Messaoudi and B. Said Houari, Blow up of solutions of a class of wave equations with nonlinear damping and source terms, Math. Methods Appl. Sci., 27 (2004), 1687-1696.  doi: 10.1002/mma.522.

[21]

S. A. Messaoudi and N.-e. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nolinear Anal., 68 (2008), 785-793.  doi: 10.1016/j.na.2006.11.036.

[22]

C. X. Miao, The time space estimates and scattering at low energy for nonlinear higher order wave equations, Math. Sin. Ser. A, 38 (1995), 708-717. 

[23]

J. E. Munoz RiveraE. C. Lapa and R. Baretto, Decay rates for viscolastic plates with memory, J. Elasticity, 44 (1996), 61-87.  doi: 10.1007/BF00042192.

[24]

H. Pecher, Die existenz reguläer Lösungen für Cauchy-und anfangs-randwertproble-me michtlinear wellengleichungen, Mathematische Zeitschrift, 140 (1974), 263-279.  doi: 10.1007/BF01214167.

[25]

F. Tahamatani and M. Shahrouzi, General existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source, Bound. Value Probl., 2012 (2012), 50, 15 pp. doi: 10.1186/1687-2770-2012-50.

[26]

G. Todorova, Cauchy problem for a nonlinear wave with nonlinear damping and source terms, C. R. Acad. Sci. Paris Ser. I, 326 (1998), 191-196.  doi: 10.1016/S0764-4442(97)89469-4.

[27]

G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.

[28]

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

[29]

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

[30]

B. Wang, Nonlinear scattering theory for a class of wave equations in $H^{s}$, J. Math. Anal. Appl., 296 (2004), 74-96.  doi: 10.1016/j.jmaa.2004.03.050.

[31]

S.-T. Wu, Blow-up of solutions for an integro-differential equation with a nonlinear source, Elect. J. Diff. Eqs., (2006), No. 45, 9 pp.

[32]

Y. Ye, Existence and asymptotic behavior of global solutions for a class of nonlinear higher-order wave equation, Journal of Inequalities and Applications, 2010 (2010), Article number: 394859. doi: 10.1155/2010/394859.

[33]

Y. Ye, Global existence and blow-up of solutions for higher-order viscoelastic wave equation with a nonlinear source term, Nonlinear Analysis, 112 (2015), 129-146.  doi: 10.1016/j.na.2014.09.001.

[34]

Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^{n}$, Applied Math Letters, 18 (2005), 281-286.  doi: 10.1016/j.aml.2003.07.018.

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