Yıl 2023,
Cilt: 6 Sayı: 1, 32 - 41, 30.04.2023
Erhan Pişkin
,
Gülistan Butakın
Kaynakça
- [1] Z. Jiang, S. Zheng, X. Song, Blow up analysis for a nonlinear diffusion equation with nonlinear boundary conditions, Appl. Math. Lett., 17(2004),
193-199.
- [2] G. Kirchhoff, Vorlesungen ¨uber Mechanik, Teubner, Leipzig, 1883.
- [3] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383-1406.
- [4] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesque and Sobolev Spaces with Variable Exponents, Springer, 2011.
- [5] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, 2000.
- [6] X. Wu, B. Guo, W. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math.
Lett., 26(2013), 539-543
- [7] K. Baghaei, M. B. Ghaemi, M. Hesaaraki, Lower bounds for the blow-up time in a semilinear parabolic problem involving a variable source, Appl.
Math. Lett., 27(2014), 49-52.
- [8] A. Khelghati, K. Baghaei, Blow up in a semilinear parabolic problem with variable source under positive initial energy, Appl. Anal., 94(9)(2015),
1888-1896.
- [9] A. Rahmoune, B. Benabderrahmane, Bounds for blow-up time in a semilinear parabolic problem with variable exponents, Stud. Univ. Babe s-Bolyai
Math., 67(2022), 181-188.
- [10] H. Wang, Y. He, On blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math. Lett.,
26(2013), 1008-1012.
- [11] C. Qu, W. Zhou, B. Liang, Asymptotic behavior for a fourth-order parabolic equation modeling thin film growth, Appl. Math. Lett., 78(2018), 141-146.
- [12] A. Khaldi, A. Ouaoua, M. Maouni, Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable
exponents, Math. Bohem., 147(2022), 471-484.
- [13] S. Antontsev, J. Ferreira, E. Pis¸kin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities,
Electron. J. Differ. Equ., 2021(2021), 1-18.
- [14] S. A. Messaoudi, A. A. Talahmeh, Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent
nonlinearities, Discrete Contin. Dyn. Syst., 15(5)(2022), 1233-1245.
- [15] E. Pis¸kin, N. Yılmaz, Blow up of solutions for a system of strongly damped Petrovsky equations with variable exponents, Acta Univ. Apulensis, 71(2022)
87-99.
- [16] A. Rahmoune, Bounds for blow-up time in a nonlinear generalized heat equation, Appl. Anal., 101(6) (2022) 1871-1879.
- [17] M. Shahrouzi, J. Ferreira, E. Pis¸kin, N. Boumaza, Blow-up analysis for a class of plate viscoelastic p(x)-Kirchhoff type inverse source problem with
variable-exponent nonlinearities, Siberian Electron. Math. Report., 19(2) (2022), 912-934.
- [18] S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, Atlantis Studies in
Differential Equations, 2015.
- [19] E. Pis¸kin, B. Okutmus¸tur, An Introduction to Sobolev Spaces, Bentham Science, 2021.
- [20] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, 1994.
- [21] J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Gauthier-Villars, Paris, 1969.
- [22] S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC, 2004.
Existence and Decay of Solutions for a Parabolic-Type Kirchhoff Equation with Variable Exponents
Yıl 2023,
Cilt: 6 Sayı: 1, 32 - 41, 30.04.2023
Erhan Pişkin
,
Gülistan Butakın
Öz
This paper deals with a parabolic-type Kirchhoff equation with variable exponents. Firstly, we obtain the global existence of solutions by Faedo-Galerkin method. Later, we prove the decay of solutions by Komornik's inequality.
Kaynakça
- [1] Z. Jiang, S. Zheng, X. Song, Blow up analysis for a nonlinear diffusion equation with nonlinear boundary conditions, Appl. Math. Lett., 17(2004),
193-199.
- [2] G. Kirchhoff, Vorlesungen ¨uber Mechanik, Teubner, Leipzig, 1883.
- [3] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383-1406.
- [4] L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesque and Sobolev Spaces with Variable Exponents, Springer, 2011.
- [5] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Springer, 2000.
- [6] X. Wu, B. Guo, W. Gao, Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math.
Lett., 26(2013), 539-543
- [7] K. Baghaei, M. B. Ghaemi, M. Hesaaraki, Lower bounds for the blow-up time in a semilinear parabolic problem involving a variable source, Appl.
Math. Lett., 27(2014), 49-52.
- [8] A. Khelghati, K. Baghaei, Blow up in a semilinear parabolic problem with variable source under positive initial energy, Appl. Anal., 94(9)(2015),
1888-1896.
- [9] A. Rahmoune, B. Benabderrahmane, Bounds for blow-up time in a semilinear parabolic problem with variable exponents, Stud. Univ. Babe s-Bolyai
Math., 67(2022), 181-188.
- [10] H. Wang, Y. He, On blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy, Appl. Math. Lett.,
26(2013), 1008-1012.
- [11] C. Qu, W. Zhou, B. Liang, Asymptotic behavior for a fourth-order parabolic equation modeling thin film growth, Appl. Math. Lett., 78(2018), 141-146.
- [12] A. Khaldi, A. Ouaoua, M. Maouni, Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable
exponents, Math. Bohem., 147(2022), 471-484.
- [13] S. Antontsev, J. Ferreira, E. Pis¸kin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities,
Electron. J. Differ. Equ., 2021(2021), 1-18.
- [14] S. A. Messaoudi, A. A. Talahmeh, Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent
nonlinearities, Discrete Contin. Dyn. Syst., 15(5)(2022), 1233-1245.
- [15] E. Pis¸kin, N. Yılmaz, Blow up of solutions for a system of strongly damped Petrovsky equations with variable exponents, Acta Univ. Apulensis, 71(2022)
87-99.
- [16] A. Rahmoune, Bounds for blow-up time in a nonlinear generalized heat equation, Appl. Anal., 101(6) (2022) 1871-1879.
- [17] M. Shahrouzi, J. Ferreira, E. Pis¸kin, N. Boumaza, Blow-up analysis for a class of plate viscoelastic p(x)-Kirchhoff type inverse source problem with
variable-exponent nonlinearities, Siberian Electron. Math. Report., 19(2) (2022), 912-934.
- [18] S. Antontsev, S. Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blow-up, Atlantis Studies in
Differential Equations, 2015.
- [19] E. Pis¸kin, B. Okutmus¸tur, An Introduction to Sobolev Spaces, Bentham Science, 2021.
- [20] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Wiley, 1994.
- [21] J. L. Lions, Quelques Methodes de Resolution des Problemes Aux Limites non Lineaires, Gauthier-Villars, Paris, 1969.
- [22] S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC, 2004.