Exponential growth of solutions with Lₚ-norm of a nonlinear viscoelastic wave equation with strong damping and source and delay terms
DOI:
https://doi.org/10.24193/subbmath.2023.2.12Keywords:
: Strong damping, viscoelasticity, nonlinear source, exponential growth, delay.Abstract
In this work, we are concerned with a problem for a viscoelastic wave equation with strong damping, nonlinear source and delay terms. We show the exponential growth of solutions with Lₚ-norm. i.e.
Mathematics Subject Classification (2010): 35L05, 35L20, 58G16, 93D20.
Received 18 May 2020; Accepted 06 July 2020.
References
Ball, J., Remarks on blow-up and nonexistence theorems for nonlinear evolutions equation, Quarterly Journal of Mathematics, 28(1977), 473-486.
Berrimi, S., Messaoudi, S., Existence and decay of solutions of a viscoelastic equation with a nonlinear source, Nonlinear Analysis, 64 2006), 2314-2331.
Bialynicki-Birula, I., Mycielsk, J., Wave equations with logarithmic nonlinearities, Bull Acad Polon Sci Ser Sci. Math. Astron Phys., 23(1975), 461-466.
Cavalcanti, M.M., Cavalcanti, D., Ferreira, J., Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci., 24(2001), 1043-1053.
Cavalcanti, M.M., Cavalcanti, D., Filho, P.J.S., Soriano, J.A., Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential and Integral Equations, 14(2001), 85-116.
Cazenave, T., Haraux, A., Equation de Schrödinger avec non-linéarité logarithmique, C.R. Acad Sci. Paris Ser. A-B., 288(1979), A253-A256.
Cazenave, T., Haraux, A., Equations d’évolution avec non-linéarité logarithmique, Ann Fac Sci Toulouse Math., 2(1980), no. 1, 21-51.
Han, X., Global existence of weak solution for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50(2013), 275-283.
Haraux, A., Zuazua, E., Decay estimates for some semilinear damped hyperbolic problems, Archive for Rational Mechanics and Analysis, 100(1988), 191-206.
Georgiev, V., Todorova, G., Existence of solutions of the wave equation with nonlinear damping and source terms, Journal of Differential Equations, 109(1994), 295-308.
Gorka, P., Logarithmic quantum mechanics: Existence of the ground state, Found. Phys. Lett., 19(2006), 591-601.
Gorka, P., Convergence of logarithmic quantum mechanics to the linear one, Lett. Math. Phys., 81(2007), 253-264.
Kafini, M., Messaoudi, S.A., A blow-up result in a Cauchy viscoelastic problem, Applied Mathematics Letters, 21(2008),549-553.
Kafini, M., Messaoudi, S.A., A blow-up result in a Cauchy viscoelastic problem, Applied Mathematics Letters, 21(2008), 549-553. http://dx.doi.org/10.1016/j.aml.2007.07.004
Kafini, M., Messaoudi, S.A., Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay, Applicable Analysis, (2018). DOI: 10.1080/00036811.2018.1504029.
Levine, H.A., Instability and nonexistence of global solutions of nonlinear wave equation of the form P utt = Au + F (u), Transactions of the American Mathematical Society, 192(1974), 1-21.
Liang, G., Zhaoqin, Y., Guonguang, L., Blow up and global existence for a nonlinear viscoelastic wave equation with strong damping and nonlinear damping and source terms, Applied Mathematics, 6(2015),806-816.
Messaoudi, S.A., Blow up in a nonlinearly damped wave equation, Mathematische Nachrichten, 231(2001), 105-111.
Messaoudi, S.A., Blow up and global existence in a nonlinear viscoelastic wave equation, Mathematische Nachrichten, 260(2003), 58-66. http://dx.doi.org/10.1002/mana.200310104.
Messaoudi, S.A., Blow up of positive-initial-energy Solutions of a nonlinear viscoelastic hyperbolic equation, Journal of Mathematical Analysis and Applications, 320(2006), 902- 915.
Nicaise, S., Pignotti, C., Stabilization of the wave equation with boundary or internal distributed delay, Diff. Int. Equs., 21(2008), no. 9-10, 935-958.
Shun-Tang, W., Long-Yi, T., On global existence and blow-up of solutions or an integro- differential equation with strong damping, Taiwanese J. Math., 10(2006), 979-1014.
Song, H.T., Xue, D.S., Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Analysis, 109(2014), 245-251. http://dx.doi.org/10.1016/j.na.2014.06.012.
Song, H.T., Zhong, C.K., Blow-up of solutions of a nonlinear viscoelastic wave equation, Nonlinear Analysis: Real World Applications, 11(2010), 3877-3883. http://dx.doi.org/10.1016/j.nonrwa.2010.02.015.
Zennir, K., Exponential growth of solutions with Lp-norm of a nonlinear viscoelastic hyperbolic equation, J. Nonlinear Sci. Appl., 6(2013), 252-262.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
