8d76d3cc-da53-4d6d-9d9c-5110712ab02f20250205062603968wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS2224-26781109-277710.37394/23202http://wseas.org/wseas/cms.action?id=40671220241220242310.37394/23202.2024.23https://wseas.com/journals/systems/2024.phpAbidiKhedidjaLaboratory of Pure and Applied Mathematics, Laghouat University, ALGERIASaadaouiMohamedDepartment of Technical Sciences, Laboratory of Pure and Applied Mathematics, Laghouat University, ALGERIARahmouneAbitaDepartment of Technical Sciences, Laboratory of Pure and Applied Mathematics, Laghouat University, ALGERIAIn this paper, we study a class of semilinear m(.)-Laplacian equations with variable exponent sources. By using the potential well method, we discuss this problem at three different initial energy levels. When the initial energy is sub-critical, we obtain the blow-up result and estimate the lower and upper bounds of the blow-up time. In the case of critical initial energy, we prove global existence, asymptotic behavior, and finite-time blow-up and determine the lower bound of the blow-up time. For super-critical initial energy, we establish the finite-time blow-up and estimate the lower and upper bounds of the blow-up time.123120241231202446548348https://wseas.com/journals/systems/2024/a965102-031(2024).pdf10.37394/23202.2024.23.48https://wseas.com/journals/systems/2024/a965102-031(2024).pdfC.V. Pao, Nonlinear parabolic and elliptic equations. New York and London: Plenum Press; 1992. DOI: 10.1007/978-1-4615-3034- 3. 10.1016/j.cam.2010.01.026S.N. Antonsev, S.I. Shmarev, Blow up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math., 234(2010), 2633-2645. DOI: 10.1016/j.cam.2010.01.026. 10.2977/prims/1195193108M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ. Res. Inst. Math. Sci., 8 (1972), 211-229. DOI: 10.2977/prims/1195193108. 10.1007/978-3-642-18363-8_3L. Diening, P. Harjulehto, P. Hästö, M. Rûžička, Lebesgue and Sobolev Spaces with Variable Exponents, in: Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Heidelberg, 2011. DOI: 10.1007/978-3-642- 18363-8. M. Rûžička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lectures Notes in Mathematics, vol. 1748, Springer-Verlag, Berlin, 2000. DOI: 10.1007/BFb0104029. 10.1016/j.jmaa.2007.11.046L. Songzhe, G. Wenjie, C. Chunling and Y. Hongjun, Study of the solutions to a model porous medium equation with variable exponent of nonlinearity, J. Math. Anal. Appl., 342(2008), 27-38. DOI: 10.1016/j.jmaa.2007.11.046. 10.1016/j.camwa.2008.01.017R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing, Computers and Mathematics with Applications. 56(2008), 874-882. DOI: 10.1016/j.camwa.2008.01.017. 10.1016/j.nonrwa.2011.04.015Z. Guo, Q. Liu, J. Sun and B. Wu, Reactiondiffusion systems with 𝑝(𝑥)-growth for image denoising, Nonlinear Analysis: Real World Applications, 12(2011), 2904-2918. doi.org/10.1016/j.nonrwa.2011.04.015. 10.1007/s00030-012-0153-6G. Akagi and K. Matsuura, Nonlinear diffusion equations driven by 𝑝(𝑥)-Laplacian, NoDEA Nonlinear Differential Equations Appl., 20(2013), 37-64. DOI: doi.org/10.1007/s00030-012-0153-6. J. Simsen. A Global attractor for a 𝑝(𝑥)- Laplacian problem, Nonlinear Anal., 73(2010), 3278-3283. DOI: 10.37394/23206.2023.22.51. 10.1016/j.jmaa.2012.08.047J. Simsen, M.S. Simsen and M.R.T. Primo, Continuity of the flows and upper semicontinuity of global attractors for 𝑝𝑠(𝑥)- Laplacian parabolic problems, J. Math. Anal. Appl., 398(2013), 138-150. doi.org/10.1016/j.jmaa.2012.08.047. 10.1063/1.5005100F.C. Kong, Z.G. Luo, and F.L. Chen, Solitary wave solutions for singular non-Newtonian filtration equations. J. Math. Phys., 58(9)(2017), 093506. DOI: 10.1063/1.5005100. 10.1093/imamat/hxt043Y.L. Wang, Z.Y. Xiang, The interfaces of an inhomogeneous non-Newtonian polytropic filtration equation with convection. IMA J. Appl. Math., 80(2015), 354-375. DOI: 10.1093/imamat/hxt043. J.L. Vázquez, The Porous Medium Equation: Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007. J.R. Philip, 𝑁-diffusion. Austral. J. Phys., 14 (1961), 1-13. DOI: 10.1071/PH610001. 10.1016/0022-0396(77)90196-6H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations. J. Differential Equations, 26(1977), 291-319. DOI: 10.1016/0022-0396(77)90196-6. 10.1016/j.camwa.2014.10.018A.M. Kbiri, S.A. Messaoudi, and H.B. Khenous, A blow-up result for nonlinear generalized heat equation, Comput. Math. Appl., 68(12) (2014), 1723-1732. https://doi.org/10.1016/j.camwa.2014.10.018. 10.37394/23206.2023.22.51S. Benkouider and A. Rahmoune, The Exponential Growth of Solution, Upper and Lower Bounds for the Blow-Up Time for a Viscoelastic Wave Equation with VariableExponent Nonlinearities, WSEAS Transactions on Mathematics, vol. 22, pp. 451-465, 2023, https://doi.org/10.37394/23206.2023.22.51. 10.1002/mma.3253L. Peng, Blow-up phenomena for a pseudoparabolic equation. Math Meth Appl Sci., 38(2014), 2636-2641. DOI: 10.1002/mma.3253. 10.1080/00036811.2020.1789597A, Rahmoune, Bounds for blow-up time in a nonlinear generalized heat equation. Applicable Analysis. (2020), 1871-1879. DOI: doi.org/10.1080/00036811.2020.1789597. 10.1007/s00205-002-0208-7E. Acerbi, G. Mingione, Regularity results for stationary eletrorheological fluids, Arch. Ration. Mech. Anal., 164(2002) 213-259. DOI: 10.1007/s00205-002-0208-7. 10.1007/s10114-023-1619-7P. Yue, V.D. Rădulescu, and R. Zhang XU, Global Existence and Finite Time Blow-up for the 𝑚-Laplacian Parabolic Problem. Acta Mathematica Sinica, English Series. 39, No. 8 (2023), 1497-1524. DOI: doi.org/10.1007/s10114-023-1619-7.