May  2022, 15(5): 1233-1245. doi: 10.3934/dcdss.2021107

Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities

1. 

Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, UAE

2. 

Department of Mathematics, Birzeit University, West Bank, Palestine

Received  July 2021 Revised  August 2021 Published  May 2022 Early access  September 2021

This work is concerned with a system of wave equations with variable-exponent nonlinearities acting in both equations. We, first, discuss the well-posedness then prove a blow up result for solutions with negative initial energy.

Citation: Salim A. Messaoudi, Ala A. Talahmeh. Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1233-1245. doi: 10.3934/dcdss.2021107
References:
[1]

K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270. 

[2]

S. Antontsev, Wave equation with $p(x, t)-$Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl., 3 (2011), 503-525.  doi: 10.7153/dea-03-32.

[3]

S. Antontsev, Wave equation with $p(x, t)-$Laplacian and damping term: Blow-up of solutions, Comptes Rendus Mecanique, 12 (2011), 751-755. 

[4]

S. Antontsev and J. Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., 93 (2013), 62-77.  doi: 10.1016/j.na.2013.07.019.

[5]

S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up, Atlantis Studies in Differential Equations, Atlantis Press 2015. doi: 10.2991/978-94-6239-112-3.

[6]

S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (2010), 2633-2645.  doi: 10.1016/j.cam.2010.01.026.

[7]

S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x, t)$-Laplacian type, Adv. Differential Equations, 10 (2005), 1053-1080. 

[8]

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

[9]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2), 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[10]

O. Bouhoufani, Existence and Asymptotic Behavior of Solutions of Certain Hyperbolic Coupled Systems with Variable Exponents, PhD Thesis 2021, University of Batna 2, Algeria.

[11]

O. Bouhoufani and I. Hamchi, Coupled system of nonlinear hyperbolic equations with variable-exponents: Global existence and stability, Mediterr. J. Math., 18 (2021), Paper No. 98, 2 pp. doi: 10.1007/s00009-020-01648-7.

[12]

D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267-293.  doi: 10.4064/sm-143-3-267-293.

[13]

D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Math. Nachr., 246/247 (2002), 53-67.  doi: 10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T.

[14]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[15]

Y. Gao and W. Gao, Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents, Bound. Value Probl., 2013 (2013), 208, 8 pp. doi: 10.1186/1687-2770-2013-208.

[16]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[17]

B. Guo and W. Gao, Blow-up of solutions to quasilinear hyperbolic equations with $p(x, t)-$Laplacian and positive initial energy, Comptes Rendus Mecanique, 342 (2014), 513-519. 

[18]

X. Han and M. Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source terms, Nonlinear Anal., 71 (2009), 5427-5450.  doi: 10.1016/j.na.2009.04.031.

[19]

A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206.  doi: 10.1007/BF00282203.

[20]

M. Kafini and S. A. Messaoudi, A blow up result for a viscoelastic system in $\mathbb{R}^N$, Electron. J. Differential Equations, (2007), No. 113, 7 pp.

[21]

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

[22]

M. Kbiri AlaouiS. A. Messaoudi and H. B. Khenous, A blow-up result for nonlinear generalized heat equation, Comput. Math. Appl., 68 (2014), 1723-1732.  doi: 10.1016/j.camwa.2014.10.018.

[23]

M. Kopáčková, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin., 30 (1989), 713-719. 

[24]

M. O. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), No. 119, 10 pp.

[25]

D. Lars, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev spaces with variable exponents, HBA Lect. Notes Math., 2017 (2011). doi: 10.1007/978-3-642-18363-8.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

S. A. Messaoudi and A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 96 (2017), 1509-1515.  doi: 10.1080/00036811.2016.1276170.

[32]

S. A. Messaoudi and A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40 (2017), 6976-6986.  doi: 10.1002/mma.4505.

[33]

S. A. MessaoudiA. A. Talahmeh and J. H. Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Comput. Math. Appl., 74 (2017), 3024-3041.  doi: 10.1016/j.camwa.2017.07.048.

[34] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monogr. Res. Notes Math., CRC Press, 2015.  doi: 10.1201/b18601.
[35]

B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92. 

[36]

L. SunY. Ren and W. Gao, Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources, Comput. Math. Appl., 71 (2016), 267-277.  doi: 10.1016/j.camwa.2015.11.016.

[37]

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.

[38]

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.

show all references

References:
[1]

K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270. 

[2]

S. Antontsev, Wave equation with $p(x, t)-$Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl., 3 (2011), 503-525.  doi: 10.7153/dea-03-32.

[3]

S. Antontsev, Wave equation with $p(x, t)-$Laplacian and damping term: Blow-up of solutions, Comptes Rendus Mecanique, 12 (2011), 751-755. 

[4]

S. Antontsev and J. Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal., 93 (2013), 62-77.  doi: 10.1016/j.na.2013.07.019.

[5]

S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up, Atlantis Studies in Differential Equations, Atlantis Press 2015. doi: 10.2991/978-94-6239-112-3.

[6]

S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234 (2010), 2633-2645.  doi: 10.1016/j.cam.2010.01.026.

[7]

S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of $p(x, t)$-Laplacian type, Adv. Differential Equations, 10 (2005), 1053-1080. 

[8]

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

[9]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford Ser. (2), 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.

[10]

O. Bouhoufani, Existence and Asymptotic Behavior of Solutions of Certain Hyperbolic Coupled Systems with Variable Exponents, PhD Thesis 2021, University of Batna 2, Algeria.

[11]

O. Bouhoufani and I. Hamchi, Coupled system of nonlinear hyperbolic equations with variable-exponents: Global existence and stability, Mediterr. J. Math., 18 (2021), Paper No. 98, 2 pp. doi: 10.1007/s00009-020-01648-7.

[12]

D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math., 143 (2000), 267-293.  doi: 10.4064/sm-143-3-267-293.

[13]

D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Math. Nachr., 246/247 (2002), 53-67.  doi: 10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T.

[14]

X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m, p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.  doi: 10.1006/jmaa.2000.7617.

[15]

Y. Gao and W. Gao, Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents, Bound. Value Probl., 2013 (2013), 208, 8 pp. doi: 10.1186/1687-2770-2013-208.

[16]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[17]

B. Guo and W. Gao, Blow-up of solutions to quasilinear hyperbolic equations with $p(x, t)-$Laplacian and positive initial energy, Comptes Rendus Mecanique, 342 (2014), 513-519. 

[18]

X. Han and M. Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source terms, Nonlinear Anal., 71 (2009), 5427-5450.  doi: 10.1016/j.na.2009.04.031.

[19]

A. Haraux and E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal., 100 (1988), 191-206.  doi: 10.1007/BF00282203.

[20]

M. Kafini and S. A. Messaoudi, A blow up result for a viscoelastic system in $\mathbb{R}^N$, Electron. J. Differential Equations, (2007), No. 113, 7 pp.

[21]

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

[22]

M. Kbiri AlaouiS. A. Messaoudi and H. B. Khenous, A blow-up result for nonlinear generalized heat equation, Comput. Math. Appl., 68 (2014), 1723-1732.  doi: 10.1016/j.camwa.2014.10.018.

[23]

M. Kopáčková, Remarks on bounded solutions of a semilinear dissipative hyperbolic equation, Comment. Math. Univ. Carolin., 30 (1989), 713-719. 

[24]

M. O. Korpusov, Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), No. 119, 10 pp.

[25]

D. Lars, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev spaces with variable exponents, HBA Lect. Notes Math., 2017 (2011). doi: 10.1007/978-3-642-18363-8.

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

S. A. Messaoudi and A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal., 96 (2017), 1509-1515.  doi: 10.1080/00036811.2016.1276170.

[32]

S. A. Messaoudi and A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci., 40 (2017), 6976-6986.  doi: 10.1002/mma.4505.

[33]

S. A. MessaoudiA. A. Talahmeh and J. H. Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Comput. Math. Appl., 74 (2017), 3024-3041.  doi: 10.1016/j.camwa.2017.07.048.

[34] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monogr. Res. Notes Math., CRC Press, 2015.  doi: 10.1201/b18601.
[35]

B. Said-Houari, Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms, Differential Integral Equations, 23 (2010), 79-92. 

[36]

L. SunY. Ren and W. Gao, Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources, Comput. Math. Appl., 71 (2016), 267-277.  doi: 10.1016/j.camwa.2015.11.016.

[37]

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.

[38]

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.

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