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

Mengxian Lv; Jianghao Hao

doi:10.3934/mcrf.2021058

Article Contents

2023, Volume 13, Issue 1: 303-329. Doi: 10.3934/mcrf.2021058

This issue Previous Article Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations Next Article Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

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

Mengxian Lv and
Jianghao Hao^,

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

^* Corresponding author: Jianghao Hao
^* Corresponding author: Jianghao Hao

Received: June 2021

Revised: September 2021

Early access: December 2021

Published: March 2023

This research was partially supported by Natural Science Foundation of China (grant number 11871315, 61374089), Natural Science Foundation of Shanxi Province of China (grant number 201901D111021)

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Abstract

In this paper we consider a system of viscoelastic wave equations of Kirchhoff type with dynamic boundary conditions. Supposing the relaxation functions $ g_i $ $ (i = 1, 2, \cdots, l) $ satisfy $ g_i(t)\leq-\xi_i(t)G(g_i(t)) $ where $ G $ is an increasing and convex function near the origin and $ \xi_i $ 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.

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Mathematics Subject Classification: Primary: 35B35, 93D15; Secondary: 80A17.

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