Existence and decay of solutions for a higher-order viscoelastic wave equation with logarithmic nonlinearity

Yıl 2021, Cilt: 70 Sayı: 1, 300 - 319, 30.06.2021

Erhan Pişkin , Nazlı Irkıl

https://doi.org/10.31801/cfsuasmas.718432

Cited By: 1

Öz

The main goal of this paper is to study for the local existence and decay estimates results for a high-order viscoelastic wave equation with logarithmic nonlinerity. We obtain several results: Firstly, by using Feado-Galerkin method and a logaritmic Sobolev inequality, we proved local existence of solutions. Later, we proved general decay results of solutions.

Anahtar Kelimeler

Decay, existence, logarithmic nonlinearity

Kaynakça

• Adams, R. A. and Fournier, J. J., Sobolev spaces, Elsevier, 2003.
• Al-Gharabli, M. M., Guesmia, A. and Messaoudi, S. A., Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity, Applicable Analysis 99 (1) (2020), 50-74.
• Al-Gharabli, M. M. and Messaoudi, S. A., The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term, Journal of Mathematical Analysis and Applications 454 (2) (2017), 1114-1128.
• Al-Gharabli, M. M. and Messaoudi, S. A., Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, Journal of Evolution Equations 18 (1) (2018), 105-125.
• Bartkowski, K. and Górka, P., One-dimensional Klein-Gordon equation with logarithmic nonlinearities, Journal of Physics A: Mathematical and Theoretical 41 (35) (2008), 355201.
• Bialynicki-Birula, I. and Mycielski, J., Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. Cl 3 (23) (1975), 461.
• Cazenave, T. and Haraux, A., Équations d'évolution avec non linéarité logarithmique, In Annales de la Faculté des sciences de Toulouse: Mathématiques 2 (1980), 21-51.
• De Martino, S., Falanga, M., Godano, C. and Lauro, G., Logarithmic Schrödinger-like equation as a model for magma transport, EPL (Europhysics Letters) 63 (3) (2003), 472.
• Górka, P., Logarithmic Klein-Gordon equation., Acta Physica Polonica B 40 (1) (2009).
• Gross, L., Logarithmic sobolev inequalities, American Journal of Mathematics 97 (4) (1975), 1061-1083.
• Han, X., Global existence of weak solutions for a logarithmic wave equation arising from q-ball dynamics, Bull. Korean Math. Soc 50 (1) (2013), 275-283.
• Martinez, P., A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control, Optimisation and Calculus of Variations 4 (1999), 419-444.
• Peyravi, A., General stability and exponential growth for a class of semi-linear wave equations with logarithmic source and memory terms, Applied Mathematics & Optimization 81 (2) (2020), 545-561.
• Piskin, E., Sobolev Spaces, Seçkin Publishing, 2017.
• Piskin, E. and Irkıl, N., Mathematical behavior of solutions of fourth-order hyperbolic equation with logarithmic source term, In Conference Proceedings of Science and Technology 2 (2019), 27-36.
• Piskin, E. and Irkıl, N., Well-posedness results for a sixth-order logarithmic boussinesq equation, Filomat 33 (13) (2019), 3985-4000.