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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


A Damped Nonlinear Hyperbolic Equation with Nonlinear Strain Term

Journal of Applied Nonlinear Dynamics 11(1) (2022) 171--177 | DOI:10.5890/JAND.2022.03.010

Eugenio Cabanillas Lapa

Instituto de Investigaci\'on, Facultad de Ciencias Matem\'aticas-UNMSM, Lima-Per\'u

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Abstract

In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation $u_{tt}+ u_{xxxx}+ \alpha u_{xxxxt}= f(u_{x})_{x}$, for $f(s)=|s|^{\rho}+|s|^{\sigma},\ 1<\rho,\sigma ,\ \alpha>0$. Under suitable conditions, we prove the existence of global solutions and the exponential decay of energy. The nonlinearity\ $f(s)$\ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar (1978). The exponential decay is obtained via an integral inequality introduced by Komornik (1994).

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