Main Article Content
In this paper, the blow-up of solutions for a Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p(x,t)-Laplacian operator in the presence of time delay is obtained. Under suitable conditions on g and the variable exponent of the p(x,t)-Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
- Authors keep the rights and guarantee the Journal of Innovative Applied Mathematics and Computational Sciences the right to be the first publication of the document, licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License that allows others to share the work with an acknowledgement of authorship and publication in the journal.
- Authors are allowed and encouraged to spread their work through electronic means using personal or institutional websites (institutional open archives, personal websites or professional and academic networks profiles) once the text has been published.
D. Andrade, M. A. Jorge Silva and T. F. Ma, Exponential stability for a plate equation with p-Laplacian and memory terms, Math. Methods Appl. Sci., 35(4) (2012), 417–426.
S. Antontsev and J. Ferreira, On a viscoelastic plate equation with strong damping and p(x,t)-Laplacian. Existence and uniqueness. Diff. Integ. Equ., 27(11/12) (2014), 1147–1170.
S. Antontsev, Wave equation with p(x,t)-Laplacian and damping term: Existence and blowup, Differ. Equ. Appl., 3(4) (2011), 503–525.
S. Antontsev and S. Shmarev, Anisotropic parabolic equations with variable nonlinearity, Publ. Mat., 53(2) (2009), 355–399.
L. J. An and A. Peirce, The effect of microstructure on elastic-plastic models, SIAM J. Appl. Math., 54(3) (1994), 708–730.
L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math. 55 (1995), 136–155.
M. M. Cavalcanti, V. N. Domingos Cavalcanti, I. Lasiecka and C. M. Webler, Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. Adv. Nonlinear Anal. 6(2) (2017), 121–145.
I. Chueshov and I. Lasiecka, existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff -Boussinesq models, Discrete Contin. Dyn. Syst., 15 (2006), 777-809.
I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping. RI: AMS, vol.195. Providence, 2008.
L. Diening, P. Harjulehto, P. Hästö, and M. Ruziccka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, 2011. Series: Lecture Notes in Mathematics, Undergraduate Texts in Mathematics, 1st Edition 2017.
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.
V. Georgiev and G. Todorova, existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.
M.A. Jorge Silva and T.F. Ma, On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type, IMA J. Appl. Math. 78 (2013), 1130–1146.
M. Kafini and S.A. Messaoudi, A blow-up result in a nonlinear wave equation with delay, Mediterr J. Math., 13(1) (2016), 237-247.
M. Kafini, S. AMessaoudi and S. Nicaise,A blow-up result in a nonlinear abstract evolution system with delay, Nonlinear Differ. Equ. Appl.,23:13 (2016), 1–14.
W. J. Liu and J. Yu, On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms, Nonlinear Anal., 74(6)(2011), 2175-2190.
S. A. Messaoudi, Blow-up of positive initial energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320(2) (2006), 902-915.
M. Nakao, Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity, Opusc. Math. 34(3) (2014), 569–590.
S. Nicaise and 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(5) (2006), 1561-1585.
E. Pişkin and H. Yüksekkaya, Blow up of solution for a viscoelastic wave equation with m-Laplacian and delay terms, Tbil. Math. J., SI (7) (2021), 21-32.
P. Pukach, V. Il’kiv, Z. Nytrebych and M. Vovk, On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt–Kelvin rheological model, Opusc. Math., 37(5) (2017), 735–753.
S. G. Samko, Density C0infty (Rn) in the generalized Sobolev spaces W^m,p(x) (Rn), Dokl. Akad. Nauk, 369 (1999), 451-454.
Wu and Z. Shun-Tang, Blow-Up of Solution for A Viscoelastic Wave Equation with Delay, Acta Math. Sci., 39(1) (2019), 329–338.
Z. Yang, Long-time behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Methods Appl. Sci., 32(9) (2009), 1082–1104 .
V. V. Zhikov, On Lavrentiev's effect, Dokl., Ross. Akad. Nauk, 345(1) (1995), 10–14.
V. V. Zhikov, On Lavrentiev's phenomenon, Russian J. Math. Phys., 3(2) (1995), 249-269.
V. V. Zhikov, On the density of smooth functions in Sobolev-Orlich spaces, J. Math. Sci., 132 (2006), 285-294.