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

Yıl 2022, Cilt: 5 Sayı: 1, 1 - 9, 15.03.2022
https://doi.org/10.32323/ujma.1028304

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.

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

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.

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