97068b37-59c2-44cb-be9e-2a422f81e57320230602065708216wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051919202291920222210.37394/23206.2023.22https://wseas.com/journals/mathematics/2023.phpSoufianeBenkouiderLaboratory of Pure and Applied Mathematics, University Amar Telidji, Laghouat, ALGERIAAbitaRahmouneDepartment of Technical Sciences, Laboratory of Pure and Applied Mathematics, Laghouat University ALGERIAThis paper aims to study the model of a nonlinear viscoelastic wave equation with damping and source terms involving variable-exponent nonlinearities. First, we prove that the energy grows exponentially, and thus in 𝐿p2 and 𝐿p1 norms. For the case 2 ≤ 𝑘(. ) < 𝑝(. ), we reach the exponential growth result of a blowup in finite time with positive initial energy and get the upper bound for the blow-up time. For the case 𝑘(. ) = 2, we use the concavity method to show a finite time blow-up result and get the upper bound for the blow-up time. Furthermore, for the case 𝑘(. ) ≥ 2, under some conditions on the data, we give a lower bound for the blow-up time when the blow-up occurs.62202362202345146551https://wseas.com/journals/mathematics/2023/b045106-021(2023).pdf10.37394/23206.2023.22.51https://wseas.com/journals/mathematics/2023/b045106-021(2023).pdf10.1016/j.camwa.2008.01.017R. Aboulaich, D. Meskine and A. Souissi. New diffusion models in image processing. Comput. Math. Appl., 56.4,2008, 874–882. 10.1063/1.5089879R. Abita. Existence and asymptotic stability for the semilinear wave equation with variable-exponent nonlinearities. J. Math. Phys. 60, 122701 (2019). 10.1080/00036811.2022.2078716R. Abita. Lower and upper bounds for the blow-up time to a viscoelastic Petrovsky wave equation with variable sources and memory term. Applicable Analysis 2022. 10.1080/00036811.2021.1947494R. Abita. Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source. Applicable Analysis 2021. 10.1007/s00205-002-0208-7E, Acerbi, G, Mingione. Regularity results for stationary eletrorheological fluids. Arch. Ration. Mech. Anal, 164, 2002, 213–259. 10.1007/s00526-008-0188-zO. Claudianor, O. Alves and M. M. Cavalcanti. On existence, uniform decay rates and blow up for solutions of the 2-d wave equation with exponential source. Calculus of Variations and Partial Differential Equations, 34, 03, 2009. 10.1016/j.cam.2010.01.026S.N. Antonsev. Blow up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math, 234, 2010, 2633–2645. 10.1016/s1874-5733(06)80005-7S.N. Antontsev and S. I. Shmarev. Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, HandBook of Differential Equations, Stationary Partial Differential Equations, volume 3. 2006. 10.1016/j.cam.2010.01.026S.N. Antontsev and S. I. Shmarev. Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math., 234(9), 2010, 2633–2645. 10.57262/ade/1355867817S.N. Antontsev and V. Zhikov. Higher integrability for parabolic equations of 𝑝(𝑥, 𝑡)-laplacian type. Adv. Differential Equations, 10(9), 2009, 1053–1080. 10.1137/050624522Y. Chen, S. Levine and M. Rao. Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math., 66, 2006, 1383–1406. 10.1007/bf00251609C.M. Dafermos. Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal., 37, 1970, 297–308. 10.1007/978-3-642-18363-8_3L. Diening, P. Hästo, P. Harjulehto and M. Ružicka. Lebesgue and Sobolev Spaces with Variable Exponents, volume 2017. in: Springer Lecture Notes, Springer-Verlag, Berlin, 2011. 10.1007/bf01109723V. Kalantarov and O.A Ladyzhenskaya. The occurence of collapse for quasilinear equation of paprabolic and hyperbolic types. J. Sov. Math., 10, 1978, 53–70. O. Kovàcik and J. Rákosnik. On spaces 𝐿 ௣(௫)(𝛺), and 𝑊ଵ,௣(௫)(𝛺), volume 41. 1991. 10.1137/1.9781611970463.ch1L.E. Payne. Improperly posed problems in partial differential equations. Regional Conference Series in Applied Mathematics., 1975, pages 1–61. 10.1016/j.nonrwa.2010.02.015H. Song and C. Zhong. Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Analysis: Real World Applications, 11, 2010, 3877–3883. 10.1016/s0764-4442(97)89469-4G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. C.R. Acad. Sci. Paris Sér. I Math., 326, 1998, 191–196. 10.1016/j.aml.2009.01.052Y. Wang. A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Applied Mathematics Letters, 22 (2009) 1394–1400.