Wave Equation with Logarithmic Nonlinearities in Kirchhoff Type

Authors

Khaled Zennir, Department Of Mathematics, College Of Sciences and Arts, Al-Rass, Qassim University, Kingdom of Saudi Arabia.Follow
Salah Boulaaras, Department Of Mathematics, College Of Sciences and Arts, Al-Rass, Qassim University, Kingdom of Saudi Arabia.\\ Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella. Algeria.Follow
Mohamed Haiour, LANOS, Department Of Mathematics, College Of Sciences, Badji Mokhtar University, Annaba , Algeria.Follow
Mohsin Bayoud, LANOS, Department Of Mathematics, College Of Sciences, Badji Mokhtar University, Annaba , Algeria.Follow

Author Country (or Countries)

Kingdom of Saudi Arabia.

Abstract

In this paper, we study a viscoelastic wave equations of the Kirchhoff type u′′ −f (x) M(k?xuk22 )Dxu− Z t 0 g(t −s)Dxu(s)ds = auln |u|k (1) defined in any spaces dimension. It is well known that from a class of nonlinearities the logarithmic nonlinearity is distinguished by several interesting physical properties. We use weighted spaces to establish the long-time behavior of solution of (1). Furthermore, under convenient hypotheses on g and the initial data, the local-in-time existence of solution is established.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/100618

Recommended Citation

Zennir, Khaled; Boulaaras, Salah; Haiour, Mohamed; and Bayoud, Mohsin (2016) "Wave Equation with Logarithmic Nonlinearities in Kirchhoff Type," Applied Mathematics & Information Sciences: Vol. 10: Iss. 6, Article 18.
DOI: http://dx.doi.org/10.18576/amis/100618
Available at: https://digitalcommons.aaru.edu.jo/amis/vol10/iss6/18