پیوندهای مفید
آمار
| تعداد نشریات | 21 |
| تعداد شمارهها | 679 |
| تعداد مقالات | 9,894 |
| تعداد مشاهده مقاله | 70,080,629 |
| تعداد دریافت فایل اصل مقاله | 49,376,491 |
Blow up of solutions for a r(x)-Laplacian Lam\'{e} equation with variable-exponent nonlinearities and arbitrary initial energy level | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 34، دوره 13، شماره 1، خرداد 2022، صفحه 441-450 اصل مقاله (364.81 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.23671.2578 | ||
| نویسنده | ||
| Mohammad Shahrouzi* | ||
| Department of Mathematics, Jahrom University, Jahrom, Iran | ||
| تاریخ دریافت: 22 خرداد 1400، تاریخ بازنگری: 01 شهریور 1400، تاریخ پذیرش: 24 شهریور 1400 | ||
| چکیده | ||
| In this paper, we consider the nonlinear $r(x)-$Laplacian Lam'{e} equation $$ u_{tt}-\Delta_{e}u-div\big(|\nabla u|^{r(x)-2}\nabla u\big)+|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u $$ in a smoothly bounded domain $\Omega\subseteq R^{n},\ n\geq1$, where $r(.),\ m(.)$ and $p(.)$ are continuous and measurable functions. Under suitable conditions on variable exponents and initial data, the blow-up of solutions is proved with negative initial energy as well as positive. | ||
| کلیدواژهها | ||
| blow-up؛ variable-exponent nonlinearities؛ elasticity operator؛ arbitrary initial energy | ||
| مراجع | ||
|
[1] S. Antontsev, Wave equation with p(x, t)-Laplacian and damping term: blow-up of solutions, CR Mechanique, 339(12), (2011) 751–755. [2] S. Antontsev, J. Ferreira and E. Pi¸skin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities, Electron. J. Differ. Equ., 2021(6), (2021) 1–18. [3] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Q. J. Math., 28(4), (1977) 473–486. [4] A. Beniani, N. Taouaf and A. Benaissa, Well-posedness and exponential stability for coupled Lame´ system with viscoelastic term and strong damping, Comput. Math. Appl., 75, (2018) 4397–4404. [5] A. Bchatnia, A. Guesmia, Well posedness and asymptotic stability for the Lam´e system with infinite memories in a bounded domain, Math. Control Relat. Fields., 4, (2014) 451–463. [6] S. Boulaaras, Well-posedness and exponential decay of solutions for a coupled Lam´e system with viscoelastic term and logarithmic source term, Appl. Anal., DOI: 10.1080/00036811.2019.1648793 (2019). [7] L. Diening, H. Petteri, P. Hasto, et al., Lebesgue and Sobolev spaces with variable exponents, In Lecture Note Mathematics, Vol. 2017 (2011). [8] D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent, Stud. Math., 143, (2000) 267–293. [9] D. Edmunds, J. Rakosnik, Sobolev embeddings with variable exponent II, Math. Nachr., 246, (2002) 53–67. [10] X. Fan and D. Zhao, On the spaces L p(x) and W m,p(x) (Ω), J. Math. Anal. Appl., 263, (2001) 424–446. [11] X. Fan and Q. H. Zhang, Existence of solutions for p(x)−Laplacian Dirichlet problem, Nonlinear Anal., 52, (2003) 1843–1852. [12] V. Georgiev, G. Todorova, Existence of solutions to the wave equation with nonlinear damping and source terms, J. Diff. Eqns., 109(2), (1994) 295–308. [13] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5(1), (1974) 138–146. [14] F. Li, G. Y. Gao, Uniform stability of the solution for a memory-type elasticity system with nonhomogeneous boundary control condition, J. Dynam. and Cont. Systems., 23, (2017) 301–315. [15] S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr., 231(1), (2001) 1–7. [16] S. A. Messaoudi, A. A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Math. Meth. Appl. Sci., (2017) 1–11. [17] S. A. Messaoudi, A. A. Talahmeh, On wave equation: review and recent results, Arab. J. Math., 7, (2018) 113–145. [18] E. Pi¸skin, Global nonexistence of solutions for a nonlinear Klein-Gordon equation with variable exponents, Appl. Math. E-Notes, 19, (2019) 315–323. [19] E. Pi¸skin, A. Fidan, Blow up of solutions for viscoelastic wave equations of Kirchhoff type with arbitrary positiveئ initial energy, Electron. J. Differential Equations, 2017 (242), (2017) 1–10. [20] H. Song, Blow up of arbitrary positive initial energy solutions for a viscoelastic wave equation, Nonlinear Anal. Real World Appl. 26, (2015) 306–314. [21] M. Shahrouzi, On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities, Comp. Math. Appl., 75, (2018) 3946–3956. [22] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal., 149(2), (1999) 155–182. | ||
|
آمار تعداد مشاهده مقاله: 16,137 تعداد دریافت فایل اصل مقاله: 7,723 |
||