General Decay Rates for a Viscoelastic Wave Equation with Dynamic Boundary Conditions and Past History

2018 ◽  
Vol 15 (3) ◽  
Author(s):  
Baowei Feng
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Past History ◽  
Decay Rates ◽  
General Decay ◽  
Dynamic Boundary Conditions ◽  
Viscoelastic Wave Equation ◽  
Dynamic Boundary ◽  
Viscoelastic Wave

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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mengxian Lv ◽  
Jianghao Hao
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Energy ◽  
Decay Rates ◽  
General Decay ◽  
Dynamic Boundary Conditions ◽  
Positive Initial Energy ◽  
Relaxation Functions ◽  
Dynamic Boundary

<p style='text-indent:20px;'>In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions <inline-formula><tex-math id="M1">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (i = 1, 2, \cdots, l) $\end{document}</tex-math></inline-formula> satisfy <inline-formula><tex-math id="M3">\begin{document}$ g_i(t)\leq-\xi_i(t)G(g_i(t)) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is an increasing and convex function near the origin and <inline-formula><tex-math id="M5">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> 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.</p>

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