# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021058
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## General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions

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

* Corresponding author: Jianghao Hao

Received  June 2021 Revised  September 2021 Early access December 2021

Fund Project: 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)

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.

Citation: Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021058
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##### References:
 [1] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [2] Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671 [3] Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669 [4] Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125 [5] Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 [6] Lan Qiao, Sining Zheng. Non-simultaneous blow-up for heat equations with positive-negative sources and coupled boundary flux. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1113-1129. doi: 10.3934/cpaa.2007.6.1113 [7] Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020 [8] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399 [9] Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130 [10] Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63 [11] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [12] Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011 [13] Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2687-2707. doi: 10.3934/cpaa.2020239 [14] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [15] Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 [16] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [17] Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333 [18] Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 [19] Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 [20] V. Casarino, K.-J. Engel, G. Nickel, S. Piazzera. Decoupling techniques for wave equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 761-772. doi: 10.3934/dcds.2005.12.761

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