General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

Abstract

In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions \begin{document}$ g_i $\end{document} \begin{document}$ (i = 1, 2, \cdots, l) $\end{document} satisfy \begin{document}$ g_i(t)\leq-\xi_i(t)G(g_i(t)) $\end{document} where \begin{document}$ G $\end{document} is an increasing and convex function near the origin and \begin{document}$ \xi_i $\end{document} are nonincreasing, we establish some optimal and general decay rates of the energy using the multiplier method and some properties of convex functions. Moreover, we obtain the finite time blow-up result of solution with nonpositive or arbitrary positive initial energy. The results in this paper are obtained without imposing any growth condition on weak damping term at the origin. Our results improve and generalize several earlier related results in the literature.