Global well-posedness for a class of Kirchhoff-type wave system
type: Item
instance of: scholarly article
title: Global well-posedness for a class of Kirchhoff-type wave system
zbMATH Open document ID: 1415.35183
DOI: 10.1155/2017/1620417
author: Xiaoli Jiang
author: Xiao-Feng Wang
published in: Advances in Mathematical Physics
publication date: 2018-10-23T00:00:00Z
review text: Summary: In recent years, the small initial boundary value problem of the Kirchhoff-type wave system attracts many scholars' attention. However, the big initial boundary value problem is also a topic of theoretical significance. In this paper, we devote oneself to the well-posedness of the Kirchhoff-type wave system under the big initial boundary conditions. Combining the potential well method with an improved convex method, we establish a criterion for the well-posedness of the system with nonlinear source and dissipative and viscoelastic terms. Based on the criteria, the energy of the system is divided into different levels. For the subcritical case, we prove that there exist the global solutions when the initial value belongs to the stable set, while the finite time blow-up occurs when the initial value belongs to the unstable set. For the supercritical case, we show that the corresponding solution blows up in a finite time if the initial value satisfies some given conditions.
Mathematics Subject Classification ID: 35L53
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6962590
zbMATH Keywords: potential well method
zbMATH Keywords: improved convex method
zbMATH Keywords: nonlinear source
zbMATH Keywords: dissipative and viscoelastic terms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/1620417
OpenAlex ID: W2772189529
cites work: Nonexistence of local solutions to semilinear partial differential inequalities
cites work: On a quasilinear wave equation with memory
cites work: Q4299985
cites work: On global existence and blow-up of solutions for an integro-differential equation with strong damping
cites work: Exponential energy decay of solutions for an integro-differential equation with strong damping
cites work: Existence of a solution of the wave equation with nonlinear damping and source terms
cites work: Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Pu tt = -Au + ℱ(u)
cites work: Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations
cites work: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation
cites work: Global nonexistence theorems for a class of evolution equations with dissipation
cites work: Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation
cites work: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings
cites work: Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation
cites work: Q5426760
cites work: Finite time blow-up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level
cites work: Abstract non linear wave equations
cites work: Systems of nonlinear wave equations with damping and source terms.
cites work: On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms
cites work: Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms.
cites work: Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms
cites work: Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations
cites work: Critically and degenerately damped systems of nonlinear wave equations with source terms
cites work: Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source
cites work: Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source
cites work: General decay of solutions of a nonlinear system of viscoelastic wave equations
cites work: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms
cites work: Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms
cites work: Global existence and uniform decay of solutions for a system of wave equations with dispersive and dissipative terms
cites work: Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms
cites work: Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping
cites work: Uniform stabilization of solutions of a nonlinear system of viscoelastic equations
cites work: Q4530278
cites work: Global existence and blow up of solutions for some hyperbolic systems with damping and source terms
cites work: Global existence and decay of solutions of a nonlinear system of wave equations
cites work: On decay and blow-up of solutions for a system of nonlinear wave equations
cites work: A difference inequality and its application to nonlinear evolution equations
cites work: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy
cites work: A blow up result for viscoelastic equations with arbitrary positive initial energy
cites work: A blow-up result for Kirchhoff-type equations with high energy
cites work: Global well-posedness for strongly damped viscoelastic wave equation
cites work: Nonlinear wave equations and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level
cites work: Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation
description: scientific article; zbMATH DE number 6962590