Browse > Article
http://dx.doi.org/10.4134/JKMS.j210084

EXISTENCE AND GENERAL DECAY FOR A VISCOELASTIC EQUATION WITH LOGARITHMIC NONLINEARITY  

Ha, Tae Gab (Department of Mathematics and Institute of Pure and Applied Mathematics Jeonbuk National University)
Park, Sun-Hye (Office for Education Accreditation Pusan National University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1433-1448 More about this Journal
Abstract
In the present work, we investigate a viscoelastic equation involving a logarithmic nonlinear source term. After proving the existence of solutions, we establish a general decay estimate of the solution using energy estimates and theory of convex functions. This result extends and complements some previous results of [9, 21].
Keywords
General decay; logarithmic source; viscoelastic equation; convex function;
Citations & Related Records
연도 인용수 순위
  • Reference
1 L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), no. 3-4, 273-303. https://doi.org/10.1007/BF02761595   DOI
2 K.-P. Jin, J. Liang, and T.-J. Xiao, Coupled second order evolution equations with fading memory: optimal energy decay rate, J. Differential Equations 257 (2014), no. 5, 1501-1528. https://doi.org/10.1016/j.jde.2014.05.018   DOI
3 J. Barrow and P. Parsons, Inflationary models with logarithmic potentials, Phys. Rev. D 52 (1995), 5576-5587.   DOI
4 S. Boulaaras, A. Draifia, and K. Zennir, General decay of nonlinear viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping and logarithmic nonlinearity, Math. Methods Appl. Sci. 42 (2019), no. 14, 4795-4814. https://doi.org/10.1002/mma.5693   DOI
5 H. Chen, P. Luo, and G. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl. 422 (2015), no. 1, 84-98. https://doi.org/10.1016/j.jmaa.2014.08.030   DOI
6 P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Polon. B 40 (2009), no. 1, 59-66.
7 T. G. Ha and S.-H. Park, Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity, Adv. Difference Equ. 2020 (2020), Paper No. 235, 17 pp. https://doi.org/10.1186/s13662-020-02694-x   DOI
8 I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507-533.
9 M. Liao and Q. Li, A class of fourth-order parabolic equations with logarithmic nonlinearity, Taiwanese J. Math. 24 (2020), no. 4, 975-1003. https://doi.org/10.11650/tjm/190801   DOI
10 J. H. Hassan, S. A. Messaoudi, and M. Zahri, Existence and new general decay results for a viscoelastic Timoshenko system, Z. Anal. Anwend. 39 (2020), no. 2, 185-222. https://doi.org/10.4171/zaa/1657   DOI
11 L. C. Nhan and L. X. Truong, Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity, Comput. Math. Appl. 73 (2017), no. 9, 2076-2091. https://doi.org/10.1016/j.camwa.2017.02.030   DOI
12 M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. Ferreira, Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Math. Methods Appl. Sci. 24 (2001), no. 14, 1043-1053. https://doi.org/10.1002/mma.250   DOI
13 J. Y. Park and S. H. Park, General decay for quasilinear viscoelastic equations with nonlinear weak damping, J. Math. Phys. 50 (2009), no. 8, 083505, 10 pp. https://doi.org/10.1063/1.3187780   DOI
14 J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, 1969.
15 L. Ma and Z. B. Fang, Energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source, Math. Methods Appl. Sci. 41 (2018), no. 7, 2639-2653. https://doi.org/10.1002/mma.4766   DOI
16 S. A. Messaoudi and N. Tatar, Exponential and polynomial decay for a quasilinear viscoelastic equation, Nonlinear Anal. 68 (2008), no. 4, 785-793. https://doi.org/10.1016/j.na.2006.11.036   DOI
17 M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci. 41 (2018), no. 1, 192-204. https://doi.org/10.1002/mma.4604   DOI
18 J. Y. Park and S. H. Park, General decay for a nonlinear beam equation with weak dissipation, J. Math. Phys. 51 (2010), no. 7, 073508, 8 pp. https://doi.org/10.1063/1.3460321   DOI
19 S. A. Messaoudi, General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), no. 2, 1457-1467. https://doi.org/10.1016/j.jmaa.2007.11.048   DOI
20 V. I. Arnold, Mathematical Methods of Classical Mechanics, translated from the Russian by K. Vogtmann and A. Weinstein, second edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4757-2063-1   DOI
21 Y. Cao and C. Liu, Initial boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity, Electron. J. Differential Equations 2018 (2018), Paper No. 116, 19 pp.
22 X. Han and M. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. 70 (2009), no. 9, 3090-3098. https://doi.org/10.1016/j.na.2008.04.011   DOI
23 H. Di, Y. Shang, and Z. Song, Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Anal. Real World Appl. 51 (2020), 102968, 22 pp. https://doi.org/10.1016/j.nonrwa.2019.102968   DOI
24 S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a quasilinear viscoelastic equation, Appl. Math. Lett. 66 (2017), 16-22. https://doi.org/10.1016/j.aml.2016.11.002   DOI
25 W. Liu, D. Chen, and S. A. Messaoudi, General decay rates for one-dimensional porous-elastic system with memory: the case of non-equal wave speeds, J. Math. Anal. Appl. 482 (2020), no. 2, 123552, 17 pp. https://doi.org/10.1016/j.jmaa.2019.123552   DOI