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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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