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Blow-up of solutions for a Kirchhoff type equation with variable-exponent nonlinearities

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition.Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time.Moreover, we obtain the upper bound estimate of the blowup time.
Słowa kluczowe
Wydawca
Rocznik
Strony
97--105
Opis fizyczny
Bibliogr. 25 poz.
Twórcy
  • Department of Mathematics, Jahrom University, Jahrom, P.O.Box: 74137-66171, Iran
  • Department of Mathematics, Jahrom University, Jahrom, P.O.Box: 74137-66171, Iran
Bibliografia
  • [1] S. Antontsev, Wave equation with p(x, t)-Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl. 3 (2011), no. 4, 503-525.
  • [2] A. Benaissa and S. A. Messaoudi, Blow-up of solutions for the Kirchhoff equation of q-Laplacian type with nonlinear dissipation, Colloq. Math. 94 (2002), no. 1, 103-109.
  • [3] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
  • [4] D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent, Studia Math. 143 (2000), no. 3, 267-293.
  • [5] D. E. Edmunds and J. Rákosník, Sobolev embeddings with variable exponent. II, Math. Nachr. 246/247 (2002), 53-67.
  • [6] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), no. 2, 424-446.
  • [7] X.-L. Fan and Q.-H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), no. 8, 1843-1852.
  • [8] J. Ferreira and S. A. Messaoudi, On the general decay of a nonlinear viscoelastic plate equation with a strong damping and p (x, t)-Laplacian, Nonlinear Anal. 104 (2014), 40-49.
  • [9] R. Ikehata, A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms, Differential Integral Equations 8 (1995), no. 3, 607-616.
  • [10] V. K. Kalantarov and O. A. Ladyženskaja, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69 (1977), 77-102, 274.
  • [11] G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig, 1883.
  • [12] T. Matsuyama and R. Ikehata, On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl. 204 (1996), no. 3, 729-753.
  • [13] S. A. Messaoudi and A. A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal. 96 (2017), no. 9, 1509-1515.
  • [14] S. A. Messaoudi, A. A. Talahmeh and J. H. Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Comput. Math. Appl. 74 (2017), no. 12, 3024-3041.
  • [15] K. Ono, On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci. 20 (1997), no. 2, 151-177.
  • [16] K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl. 216 (1997), no. 1, 321-342.
  • [17] E. Pişkin, Finite time blow up of solutions for a strongly damped nonlinear Klein-Gordon equation with variable exponents, Honam Math. J. 40 (2018), no. 4, 771-783.
  • [18] M. Shahrouzi, Asymptotic stability and blowup of solutions for a class of viscoelastic inverse problem with boundary feedback, Math. Methods Appl. Sci. 39 (2016), no. 9, 2368-2379.
  • [19] M. Shahrouzi, Blow-up analysis for a class of higher-order viscoelastic inverse problem with positive initial energy and boundary feedback, Ann. Mat. Pura Appl. (4) 196 (2017), no. 5, 1877-1886.
  • [20] M. Shahrouzi, On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities, Comput. Math. Appl. 75 (2018), no. 11, 3946-3956.
  • [21] S. T. Wu and L. Y. Tsai, Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal. 65 (2006), no. 2, 243-264.
  • [22] Z. Yang, On an extensible beam equation with nonlinear damping and source terms, J. Differential Equations 254 (2013), no. 9, 3903-3927.
  • [23] Z. Yang and Z. Gong, Blow-up solutions for viscoelastic equations of Kirchhoff type with arbitrary positive initial energy, Electron. J. Differential Equations 2016 (2016), Paper No. 332.
  • [24] Y. Ye, Global nonexistence of solutions for systems of quasilinear hyperbolic equations with damping and source terms, Bound. Value Probl. 2014 (2014), Article ID 251.
  • [25] J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput. 265 (2015), 807-818.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021)
Typ dokumentu
Bibliografia
Identyfikator YADDA
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