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doi: 10.3934/dcdss.2021107
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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 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 & Continuous Dynamical Systems - S, 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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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