Existence, uniqueness, and blow-up analysis of a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

Begüm Çalışkan Desova; Mustafa Polat; Begüm Çalışkan Desova; Mustafa Polat

doi:10.3934/era.2024155

Electronic Research Archive

2024, Volume 32, Issue 5: 3363-3376. doi: 10.3934/era.2024155

Previous Article Next Article

Research article

Existence, uniqueness, and blow-up analysis of a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

Begüm Çalışkan Desova ^{1
,
,},
Mustafa Polat ²

1.
Department of Information Security Technology, Yeditepe University, Atasehir 34755, Turkey
2.
Department of Mathematics, Yeditepe University, Atasehir 34755, Turkey

Received: 09 March 2024 Revised: 28 April 2024 Accepted: 07 May 2024 Published: 24 May 2024

For this study, we investigate the existence and uniqueness of local solutions and derive a blow-up solution for a quasi-linear bi-hyperbolic equation under dynamic boundary conditions. We utilize the contraction mapping concept to demonstrate the solution's local well-posedness and employ a concavity approach to establish the blow-up result.
- wave equation,
- local existence,
- blow-up,
- contraction mapping principle
Citation: Begüm Çalışkan Desova, Mustafa Polat. Existence, uniqueness, and blow-up analysis of a quasi-linear bi-hyperbolic equation with dynamic boundary conditions[J]. Electronic Research Archive, 2024, 32(5): 3363-3376. doi: 10.3934/era.2024155

Related Papers:

Abstract

For this study, we investigate the existence and uniqueness of local solutions and derive a blow-up solution for a quasi-linear bi-hyperbolic equation under dynamic boundary conditions. We utilize the contraction mapping concept to demonstrate the solution's local well-posedness and employ a concavity approach to establish the blow-up result.

References

[1]	C. F. Vasconcellos, L. M. Teixeira, Existence, uniqueness and stabilization for a nonlinear plate system with nonlinear damping, Ann. Fac. Sci. Toulouse, 8 (1999), 173–193. https://doi.org/10.5802/afst.928 doi: 10.5802/afst.928
[2]	M. Guedda, H. Labani, Nonexistence of global solutions to a class of nonlinear wave equations with dynamic boundary conditions, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 39–46.
[3]	S. T. Wu, L. Y. Tsai, Existence and nonexistence of global solutions for a nonlinear wave equation, Taiwanese J. Math., 13 (2009), 2069–2091. https://doi.org/10.11650/twjm/1500405658 doi: 10.11650/twjm/1500405658
[4]	V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equations, 109 (1994), 295–308. https://doi.org/10.1006/jdeq.1994.1051 doi: 10.1006/jdeq.1994.1051
[5]	E. Vitillaro, On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources, J. Differ. Equations, 265 (2017), 4873–4941. https://doi.org/10.1016/j.jde.2018.06.022 doi: 10.1016/j.jde.2018.06.022
[6]	V. Bayrak, M. Can, Nonexistence of global solutions of a quasi-linear bi-hyperbolic equation with dynamic boundary conditions, Electron. J. Qual. Theory Differ. Equations, 1999 (1999), 1–10.
[7]	M. Can, S. R. Park, F. Aliyev, Nonexistence of global solutions of a quasi-linear hyperbolic equation, Math. Inequalities Appl., 1 (1998), 45–52.
[8]	H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_tt = -Au_t+F(u)$, Trans. Am. Math. Soc., 192 (1974), 1–21. https://doi.org/10.2307/1996814 doi: 10.2307/1996814
[9]	E. Vitillaro, Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources, Discrete Contin. Dyn. Syst. S, 14 (2021), 4575–4608. https://doi.org/10.3934/dcdss.2021130 doi: 10.3934/dcdss.2021130
[10]	E. Vitillaro, Global existence of the wave equation with nonlinear boundary damping and source terms, J. Differ. Equations, 186 (2002), 259–298. https://doi.org/10.1016/s0022-0396(02)00023-2 doi: 10.1016/s0022-0396(02)00023-2
[11]	O. A. Ladyzhenskaya, V. K. Kalantarov, Blow-up theorems for quasilinear parabolic and hyperbolic equations, Zap. Nauchn. SLOMI. Steklov, 69 (1977), 77–102.
[12]	M. O. Korpusov, Blow-up of the solution of strongly dissipative generalized Klein-Gordon equations, Izv. Math., 77 (2013), 325–353. https://doi.org/10.1070/IM2013v077n02ABEH002638 doi: 10.1070/IM2013v077n02ABEH002638
[13]	E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill press, 1955.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)