Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay

Jorge Ferreira; Erhan Pişkin; Carlos Raposo; Mohammad Shahrouzi; Hazal Yüksekkaya

doi:10.32323/ujma.1028304

Araştırma Makalesi

Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay

Yıl 2022, Cilt: 5 Sayı: 1, 1 - 9, 15.03.2022

Jorge Ferreira Erhan Pişkin Carlos Raposo Mohammad Shahrouzi Hazal Yüksekkaya

https://doi.org/10.32323/ujma.1028304

Cited By: 1

Öz

This paper is concerned with a stability result for a Kirchhoff beam equation with variable exponents and time
delay. The exponential and polynomial stability decay are proved based on Komornik's inequality.

Anahtar Kelimeler

Kirchhoff beam equation, stability, time delay, variable exponents

Kaynakça

[1] S. Antontsev, Wave equation with p(x; t)-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751-755.
[2] S. Antontsev, Wave equation with p(x; t)-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3 (2011), 503-525.
[3] S. Antontsev, J. Ferreira, E. Pis¸kin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponents nonlinearities, Electr. j. differ. equ., 2021(6) (2021), 1-18.
[4] S. Antontsev, J. Ferreira, E. Pis¸kin, S. M. Siqueira Cordeiro, Existence and non-existence of solutions for Timoshenko-type equations with variable exponents, Nonlinear Anal. Real World Appl., 61 (2021), Article ID 103341.
[5] A. Antontsev, J. Ferreira, E. Pis¸kin, H. Y¨uksekkaya, M. Shahrouzi, Blow up and asymptotic behavior of solutions for a p(x)-Laplacian equation with delay term and variable exponents, Electron. J. Differ. Equ., 2021(84), (2021), 1-20.
[6] J. M. Ball, Initial boundary value problem for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.
[7] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
[8] L. Diening, P. Hasto, P. Harjulehto, M. M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
[9] X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spacesWk;p(x) (W) , J. Math. Anal. Appl., 263 (2001), 749-760.
[10] B. Feng, H. Li, Energy decay for a viscoelastic Kirchhoff plate equation with a delay term, Bound. Value Probl., 174 (2016), https://doi.org/10.1186/s13661-016-0682-8
[11] J. R. Kang, Global nonexistence of solutions for von Karman equations with variable exponents, Appl. Math. Lett., 86 (2018), 249-255.
[12] M. Kafini, S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
[13] G. Kirchhoff, Vorlesungen ¨uber mechanik, B. G. Teubner, Leipzig, 1897.
[14] O. Kovacik, J. Rakosnik, On spaces Lp(x) (W) ; andWk;p(x) (W) , Czech. Math. J., 41(116) (1991), 592-618.
[15] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson and Wiley, 1994.
[16] S. A. Messaoudi, M. Kafini, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122 (2019), 49-70.
[17] J. Musielak, Orlicz spaces and modular spaces, Springer-Verlag, Berlin, 1983.
[18] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 18, (1959) 49-65.
[19] H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Tokyo, 1950.
[20] S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
[21] S. H. Park, J. R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci., 42 (2019), 2083-2097.
[22] E. Pis¸kin, Finite time blow up of solutions of the Kirchhoff-type equation with variable exponents, Int. J. Nonlinear Anal. Appl., 11(1) (2020), 37-45.
[23] E. Pis¸kin, H. Yüksekkaya, Decay of solutions for a nonlinear Petrovsky equation with delay term and variable exponents, The Aligarh Bull. of Maths., 39(2) (2020), 63-78.
[24] H. Yüksekkaya, E. Pis¸kin, S.M. Boulaaras, B. B. Cherif, Existence, decay and blow-up of solutions for a higher-order kirchhoff-type equation with delay term, J. Funct. Spaces, 2021 (2021), Article ID 4414545
[25] H. Y¨uksekkaya, E. Pis¸kin, S. M. Boulaaras, B. B. Cherif, S. A. Zubair, Existence, nonexistence, and stability of solutions for a delayed plate equation with the logarithmic source, Adv. Math. Phys., 2021 (2021), 1-11.
[26] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, 2000.
[27] M. L. Santos, J. Ferreira, C. A. Raposo, Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary, Abstr. Appl. Anal., 8 (2005), 901-919.
[28] M. Shahrouzi, On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities, Comput. Math. with Appl., 75(11) (2018), 3946-3956.
[29] M. Tucsnak, Semi-internal stabilization for a nonlinear Euler-Bernoulli equation, Math. Method. Appl. Sci., 19 (1996), 897-907.
[30] S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

Yıl 2022, Cilt: 5 Sayı: 1, 1 - 9, 15.03.2022

Jorge Ferreira Erhan Pişkin Carlos Raposo Mohammad Shahrouzi Hazal Yüksekkaya

https://doi.org/10.32323/ujma.1028304

Cited By: 1

Öz

Kaynakça

[1] S. Antontsev, Wave equation with p(x; t)-Laplacian and damping term: blow-up of solutions, C. R. Mecanique, 339 (2011), 751-755.
[2] S. Antontsev, Wave equation with p(x; t)-Laplacian and damping term: existence and blow-up, Differential Equations Appl., 3 (2011), 503-525.
[3] S. Antontsev, J. Ferreira, E. Pis¸kin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponents nonlinearities, Electr. j. differ. equ., 2021(6) (2021), 1-18.
[4] S. Antontsev, J. Ferreira, E. Pis¸kin, S. M. Siqueira Cordeiro, Existence and non-existence of solutions for Timoshenko-type equations with variable exponents, Nonlinear Anal. Real World Appl., 61 (2021), Article ID 103341.
[5] A. Antontsev, J. Ferreira, E. Pis¸kin, H. Y¨uksekkaya, M. Shahrouzi, Blow up and asymptotic behavior of solutions for a p(x)-Laplacian equation with delay term and variable exponents, Electron. J. Differ. Equ., 2021(84), (2021), 1-20.
[6] J. M. Ball, Initial boundary value problem for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61-90.
[7] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
[8] L. Diening, P. Hasto, P. Harjulehto, M. M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, 2011.
[9] X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spacesWk;p(x) (W) , J. Math. Anal. Appl., 263 (2001), 749-760.
[10] B. Feng, H. Li, Energy decay for a viscoelastic Kirchhoff plate equation with a delay term, Bound. Value Probl., 174 (2016), https://doi.org/10.1186/s13661-016-0682-8
[11] J. R. Kang, Global nonexistence of solutions for von Karman equations with variable exponents, Appl. Math. Lett., 86 (2018), 249-255.
[12] M. Kafini, S. A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr. J. Math., 13 (2016), 237-247.
[13] G. Kirchhoff, Vorlesungen ¨uber mechanik, B. G. Teubner, Leipzig, 1897.
[14] O. Kovacik, J. Rakosnik, On spaces Lp(x) (W) ; andWk;p(x) (W) , Czech. Math. J., 41(116) (1991), 592-618.
[15] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson and Wiley, 1994.
[16] S. A. Messaoudi, M. Kafini, On the decay and global nonexistence of solutions to a damped wave equation with variable-exponent nonlinearity and delay, Ann. Pol. Math., 122 (2019), 49-70.
[17] J. Musielak, Orlicz spaces and modular spaces, Springer-Verlag, Berlin, 1983.
[18] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 18, (1959) 49-65.
[19] H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Tokyo, 1950.
[20] S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
[21] S. H. Park, J. R. Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci., 42 (2019), 2083-2097.
[22] E. Pis¸kin, Finite time blow up of solutions of the Kirchhoff-type equation with variable exponents, Int. J. Nonlinear Anal. Appl., 11(1) (2020), 37-45.
[23] E. Pis¸kin, H. Yüksekkaya, Decay of solutions for a nonlinear Petrovsky equation with delay term and variable exponents, The Aligarh Bull. of Maths., 39(2) (2020), 63-78.
[24] H. Yüksekkaya, E. Pis¸kin, S.M. Boulaaras, B. B. Cherif, Existence, decay and blow-up of solutions for a higher-order kirchhoff-type equation with delay term, J. Funct. Spaces, 2021 (2021), Article ID 4414545
[25] H. Y¨uksekkaya, E. Pis¸kin, S. M. Boulaaras, B. B. Cherif, S. A. Zubair, Existence, nonexistence, and stability of solutions for a delayed plate equation with the logarithmic source, Adv. Math. Phys., 2021 (2021), 1-11.
[26] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, 2000.
[27] M. L. Santos, J. Ferreira, C. A. Raposo, Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary, Abstr. Appl. Anal., 8 (2005), 901-919.
[28] M. Shahrouzi, On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities, Comput. Math. with Appl., 75(11) (2018), 3946-3956.
[29] M. Tucsnak, Semi-internal stabilization for a nonlinear Euler-Bernoulli equation, Math. Method. Appl. Sci., 19 (1996), 897-907.
[30] S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech., 17 (1950), 35-36.

Toplam 30 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Makaleler
Yazarlar	Jorge Ferreira 0000-0002-3209-7439 Erhan Pişkin Carlos Raposo 0000-0001-8014-7499 Mohammad Shahrouzi Hazal Yüksekkaya
Yayımlanma Tarihi	15 Mart 2022
Gönderilme Tarihi	25 Kasım 2021
Kabul Tarihi	22 Ocak 2022
Yayımlandığı Sayı	Yıl 2022 Cilt: 5 Sayı: 1

Kaynak Göster

APA	Ferreira, J., Pişkin, E., Raposo, C., Shahrouzi, M., vd. (2022). Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay. Universal Journal of Mathematics and Applications, 5(1), 1-9. https://doi.org/10.32323/ujma.1028304
AMA	Ferreira J, Pişkin E, Raposo C, Shahrouzi M, Yüksekkaya H. Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay. Univ. J. Math. Appl. Mart 2022;5(1):1-9. doi:10.32323/ujma.1028304
Chicago	Ferreira, Jorge, Erhan Pişkin, Carlos Raposo, Mohammad Shahrouzi, ve Hazal Yüksekkaya. “Stability Result for a Kirchhoff Beam Equation With Variable Exponent and Time Delay”. Universal Journal of Mathematics and Applications 5, sy. 1 (Mart 2022): 1-9. https://doi.org/10.32323/ujma.1028304.
EndNote	Ferreira J, Pişkin E, Raposo C, Shahrouzi M, Yüksekkaya H (01 Mart 2022) Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay. Universal Journal of Mathematics and Applications 5 1 1–9.
IEEE	J. Ferreira, E. Pişkin, C. Raposo, M. Shahrouzi, ve H. Yüksekkaya, “Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay”, Univ. J. Math. Appl., c. 5, sy. 1, ss. 1–9, 2022, doi: 10.32323/ujma.1028304.
ISNAD	Ferreira, Jorge vd. “Stability Result for a Kirchhoff Beam Equation With Variable Exponent and Time Delay”. Universal Journal of Mathematics and Applications 5/1 (Mart 2022), 1-9. https://doi.org/10.32323/ujma.1028304.
JAMA	Ferreira J, Pişkin E, Raposo C, Shahrouzi M, Yüksekkaya H. Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay. Univ. J. Math. Appl. 2022;5:1–9.
MLA	Ferreira, Jorge vd. “Stability Result for a Kirchhoff Beam Equation With Variable Exponent and Time Delay”. Universal Journal of Mathematics and Applications, c. 5, sy. 1, 2022, ss. 1-9, doi:10.32323/ujma.1028304.
Vancouver	Ferreira J, Pişkin E, Raposo C, Shahrouzi M, Yüksekkaya H. Stability Result for a Kirchhoff Beam Equation with Variable Exponent and Time Delay. Univ. J. Math. Appl. 2022;5(1):1-9.