Destruction of solutions for class of wave $ p(x)- $bi-Laplace equation with nonlinear dissipation

Khaled Zennir; Abderrahmane Beniani; Belhadji Bochra; Loay Alkhalifa; Khaled Zennir; Abderrahmane Beniani; Belhadji Bochra; Loay Alkhalifa

doi:10.3934/math.2023013

AIMS Mathematics

2023, Volume 8, Issue 1: 285-294. doi: 10.3934/math.2023013

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Research article

Destruction of solutions for class of wave $ p(x)- $bi-Laplace equation with nonlinear dissipation

1.
Department of Mathematics, College of Sciences and Arts, Qassim University, Ar-Rass, Saudi Arabia
2.
Laboratoire de Mathématiques Appliquées et de Modélisation, Université 8 Mai 1945 Guelma, B.P. 401, Guelma 24000, Algeria
3.
EDPs Analysis and Control Laboratory, Department of Mathematics, BP 284, University Ain Témouchent BELHADJ, Bouchaib 46000, Algeria
4.
Laboratoire de Mathématiques et Sciences Appliquées, Université de Ghardaia, BP 455, Ghardaia 47000, Algeria

Received: 03 August 2022 Revised: 15 September 2022 Accepted: 22 September 2022 Published: 28 September 2022
MSC : 35B35, 35G05, 35Q70, 45D05, 74D99

An initial value problem is considered for the nonlinear dissipative wave equation containing the $ p(x) $-bi-Laplacian operator. For this problem, sufficient conditions for the blow-up with nonpositive initial energy of a generalized solution are obtained in finite time where a wide variety of techniques are used.
- Sobolev spaces with variable exponents,
- $ p(x) $-bi-Laplace,
- blow-up,
- initial energy
Citation: Khaled Zennir, Abderrahmane Beniani, Belhadji Bochra, Loay Alkhalifa. Destruction of solutions for class of wave $ p(x)- $bi-Laplace equation with nonlinear dissipation[J]. AIMS Mathematics, 2023, 8(1): 285-294. doi: 10.3934/math.2023013

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Abstract

An initial value problem is considered for the nonlinear dissipative wave equation containing the $ p(x) $-bi-Laplacian operator. For this problem, sufficient conditions for the blow-up with nonpositive initial energy of a generalized solution are obtained in finite time where a wide variety of techniques are used.

References

[1]	R. A. Adams, Sobolev spaces, Academic press, 1975.
[2]	S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751–755. http://doi.org/10.1016/j.crme.2011.09.001 doi: 10.1016/j.crme.2011.09.001
[3]	A. Benaissa, D. Ouchenane, Kh. Zennir, Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms, Nonlinear studies, 19 (2012), 523–535.
[4]	K. Bouhali, F. Ellaggoune, Viscoelastic wave equation with logarithmic non-linearities in $\mathbb{R}^n$, J. Part. Diff. Eq., 30 (2017), 47–63. http://doi.org/10.4208/jpde.v30.n1.4 doi: 10.4208/jpde.v30.n1.4
[5]	A. Braik, Y. Miloudi, Kh. Zennir, A finite-time Blow-up result for a class of solutions with positive initial energy for coupled system of heat equations with memories, Math. Method. Appl. Sci., 41 (2018), 1674–1682. http://doi.org/10.1002/mma.4695 doi: 10.1002/mma.4695
[6]	H. Di, Y. Shang, X. Peng, Blow-up phenomena for a pseudo-parabolic equation with variable exponents, Appl. Math. Lett., 64 (2017), 67–73. http://doi.org/10.1016/j.aml.2016.08.013 doi: 10.1016/j.aml.2016.08.013
[7]	L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
[8]	A. El Khalil, M. Laghzal, D. M. Alaoui, A. Touzani, Eigenvalues for a class of singular problems involving $p(x)$-Biharmonic operator and $q(x)$-Hardy potential, Adv. Nonlinear Anal., 9 (2020), 1130–1144. http://doi.org/10.1515/anona-2020-0042 doi: 10.1515/anona-2020-0042
[9]	D. E. Edmunds, J. Rakosnik, Sobolev embedding with variable Exponent, Stud. Math., 143 (2000), 267–293. https://doi.org/10.4064/sm-143-3-267-293 doi: 10.4064/sm-143-3-267-293
[10]	X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spaces $W^{m, p(x)}(\Omega) $, J. Math. Anal. Appl., 262 (2001), 749–760. http://doi.org/10.1006/jmaa.2001.7618 doi: 10.1006/jmaa.2001.7618
[11]	Q. Gao, F. Li, Y. Wang, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Central Eur. J. Math., 9 (2011), 686–698. http://doi.org/10.2478/s11533-010-0096-2 doi: 10.2478/s11533-010-0096-2
[12]	M. K. Hamdani, N. T. Chung, D. D. Repovs, New class of sixth-order nonhomogeneous $p(x)$-Kirchhoff problems with sign-changing weight functions, Adv. Nonlinear Anal., 10 (2021), 1117–1131. http://doi.org/10.1515/anona-2020-0172 doi: 10.1515/anona-2020-0172
[13]	S. Inbo, Y.-H. Kim, Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents, Conference Publications, 2013 (2013), 695–707. http://doi.org/10.3934/proc.2013.2013.695 doi: 10.3934/proc.2013.2013.695
[14]	M. Kafini, M. I. Mustafa, A blow-up result to a delayed Cauchy viscoelastic problem, J. Integral Equ. Appl., 30 (2018), 81–94. http://doi.org/10.1216/JIE-2018-30-1-81 doi: 10.1216/JIE-2018-30-1-81
[15]	M. Liao, Non-global existence of solutions to pseudo-parabolic equations with variable exponents and positive initial energy, C. R. Mecanique, 347 (2019), 710–715. http://doi.org/10.1016/j.crme.2019.09.003 doi: 10.1016/j.crme.2019.09.003
[16]	D. Ouchenane, Kh. Zennir, M. Bayoud, Global nonexistence of solutions to system of nonlinear viscoelastic wave equations with degenerate damping and source terms, Ukr. Math. J., 65 (2013), 723–739. https://doi.org/10.1007/s11253-013-0809-3 doi: 10.1007/s11253-013-0809-3
[17]	I. D. Stircu, An existence result for quasilinear elliptic equations with variable exponents, Annals of the University of Craiova-Mathematics and Computer Science Series, 44 (2017), 299–316.
[18]	A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces, Nonlinear Anal. Theor., 69 (2008), 3629–3636. https://doi.org/10.1016/j.na.2007.10.001 doi: 10.1016/j.na.2007.10.001
[19]	Y. Zhou, A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in $\mathbb{R}^{n} $, Appl. Math. Lett., 18 (2005), 281–286. https://doi.org/10.1016/j.aml.2003.07.018 doi: 10.1016/j.aml.2003.07.018
[20]	Kh. Zennir, Growth of solutions with positive initial energy to system of degenerately damped wave equations with memory, Lobachevskii J. Math., 35 (2014), 147–156. https://doi.org/10.1134/S1995080214020139 doi: 10.1134/S1995080214020139
[21]	Kh. Zennir, T. Miyasita, Lifespan of solutions for a class of pseudo-parabolic equation with weak-memory, Alex. Eng. J., 59 (2020), 957–964. https://doi.org/10.1016/j.aej.2020.03.016 doi: 10.1016/j.aej.2020.03.016

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