Energy Decay of Solutions for a System of Higher-Order Kirchhoff Type Equations

Year 2019, Issue: 29, 89 - 100, 30.12.2019

Erhan Pişkin , Ezgi Harman

Abstract

In this work, we considered a
system of higher-order Kirchhoff type equations with initial and boundary
conditions in a bounded domain. Under suitable conditions, we proved an energy
decay result by Nakao's inequality techniques.

Keywords

Kirchhoff type equations, energy decay, damping term

Supporting Institution

Dicle University

Project Number

ZGEF.18.009

References

• G. Kirchhoff, Mechanik, Teubner, (1883).
• S. T. Wu, On Decay and Blow-Up of Solutions for a System of Nonlinear Wave Equations, Journal of Mathematical Analysis and Applications 394 (2012) 360-377.
• X. Han, M. Wang, Global Existence and Blow-Up of Solutions for a System of Nonlinear Viscoelastic Wave Equations with Damping and Source, Nonlinear Analysis 7 (2009) 5427-5450.
• S. A. Messaoudi, B. Said-Houari, Global Nonexistence of Positive Initial-Energy Solutions of a System of Nonlinear Viscoelastic Wave Equations with Damping and Source Terms, Journal of Mathematical Analysis and Applications 365 (2010) 277-287.
• B. Said-Houari, S. A. Messaoudi, A. Guesmia, General Decay of Solutions of a Nonlinear System of Viscoelastic Wave Equations, Nonlinear Differential Equations and Applications 18 (2011) 659-684.
• E. Pişkin, A Lower Bound for the Blow-Up Time of a System of Viscoelastic Wave Equations with Nonlinear Damping and Source Terms, Journal of Nonlinear Functional Analysis 2017 (2017) 1-9.
• E. Pişkin, Global Nonexistence of Solutions for a System of Viscoelastic Wave Equations with Weak Damping Terms, Malaya Journal of Matematik 3(2) (2015) 168-174.
• Y. Ye, Global Existence and Energy Decay for a Coupled System of Kirchhoff Type Equations with Damping and Source Terms, Acta Mathematicae Applicatae Sinica, English Series 32(3) (2016) 731-738.
• Q. Gao, F. Li, Y. Wang, Blow-Up of the Solution for Higher-Order Kirchhoff-Type Equations with Nonlinear Dissipation, Central European Journal of Mathematic 9(3) (2011) 686-698.
• E. Hesameddini, Y. Khalili, Blow-Up of the Solution for Higher-Order Integro-Differential Equation with Nonlinear Dissipation, Applied Mathematical Sciences 5(72) (2011) 3575-3583.
• E. Pişkin, N. Polat, Exponential Decay and Blow up of a Solution for a System of Nonlinear Higher-Order Wave Equations, American Institute of Physics Conference Proceedings 1470 (2012) 118-121.
• E. Pişkin, N. Polat, Blow Up of Positive Initial-Energy Solutions for a Coupled Nonlinear Higher-Order Hyperbolic Equations, American Institute of Physics Conference Proceedings 1676 (2015) 1-8.
• E. Pişkin, N. Polat, Global Existence and Exponential Decay of Solutions for a Class of System of Nonlinear Higher-Order Wave Equations with Strong Damping, Journal of Advanced Research in Applied Mathematics 4(4) (2012) 26-36.
• E. Pişkin, N. Polat, On the Decay of Solutions for a Nonlinear Higher-Order Kirchhoff-Type Hyperbolic Equation, Journal of Advanced Research in Applied Mathematics 5(2) (2013) 107-116.
• Y. Ye, Global Existence and Energy Decay Estimate of Solutions for a Higher-Order Kirchhoff Type Equation with Damping and Source Term, Nonlinear Analysis: Real World Applications 14 (2013) 2059-2067.
• Y. Ye, Existence and Asymptotic Behavior of Global Solutions for Aclass of Nonlinear Higher-Order Wave Equation, Journal of Inequalities and Applications 2010 (2010) 1-14.
• Y. Ye, Global Existence and Asymptotic Behavior of Solutions for a System of Higher-Order Kirchhoff-Type Equations, Electronic Journal of Qualitative Theory of Differential Equations 20 (2015) 1-12.
• J. Zhou, X. Wang, X. Song, C. Mu, Global Existence and Blowup of Solutions for a Class of Nonlinear Higher-Order Wave Equations, Zeitschrift für angewandte Mathematik und Physik 63 (2012) 461-473.
• R. A. Adams, J. J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.
• E. Pişkin, Sobolev Uzayları, Seçkin Yayıncılık (2017) (In Turkish).
• M. Nakao, A Difference Inequality and Its Application to Nonlinear Evolution Equations, Journal of the Mathematical Society of Japan 30(4) (1978) 747-762.