Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term
type: Item
instance of: scholarly article
title: Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term
zbMATH Open document ID: 1108.35121
author: Marina Ghisi
published in: Journal of Differential Equations
publication date: 2006-12-07T00:00:00Z
review text: The following Cauchy problem for the dissipative Kirchhoff equation is considered: \[ u^{\prime\prime}+\delta u^{\prime}+m(\mid A^{\frac{1}{2}}u\mid^2_H)Au=0, \] \[ u(0)=u_0,\,u^{\prime}(0)=u_1, \] where \(H\) is a Hilbert space, \(\delta>0\), \(A\) is a self-adjoint nonnegative operator in \(H\) and \(m(r)\) is a nonnegative function such that \(m(0)=0,\,m(\mid A^{\frac{1}{2}}u_0\mid^2_H)>0.\) For small initial data the existence of global solutions as well as decay of the energy as \(t\to\infty\) are proved. Earlier, see [\textit{A. T. Cousin, C. L. Frota, N. A. Lar'kin}, Differ. Integral Equ. 15, No. 10, 1219--1236 (2002; Zbl 1011.35097)], similar results were obtained for the equation \[ u_{tt}-\phi(x,t)M\left(\int_{\omega} \mid \triangledown u(s,t)\mid^2ds\right)\Delta u +g(x,t,u_t)=0. \]
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 5077857
zbMATH Keywords: Hyperbolic equations
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: small initial data
MaRDI profile type: Publication
full work available at URL: j.jde.2006.07.020
OpenAlex ID: W2043566086
cites work: Global existence for abstract evolution equations of weakly hyperbolic type
cites work: Q4385482
cites work: Nonlinear perturbations of the kirchhoff equation
cites work: Q3907929
cites work: Decay estimates for the generalized damped extensible string and beam equations
cites work: Global solutions for dissipative Kirchhoff strings with \(m(r)=r^p\) \((p<1)\)
cites work: Q4654968
cites work: The initial value problem for a stretched string
cites work: On the global solvability of Kirchhoff equation for non-analytic initial data
cites work: Q3398701
cites work: Q3480564
cites work: Q3202567
cites work: Mildly degenerate kichhoff equations with damping in unbounded domain
cites work: The Cauchy problem for Kirchhoff equations
cites work: On some quasilinear wave equations with dissipative terms
cites work: Scattering for a quasilinear hyperbolic equation of Kirchhoff type
cites work: Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three
DOI: 10.1016/J.JDE.2006.07.020
Recommended article: On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation
Recommended article: Q5267585
Recommended article: Some remarks on global solutions to nonlinear dissipative mildly degenerate Kirchhoff strings
Recommended article: Global solutions for dissipative Kirchhoff strings with \(m(r)=r^p\) \((p<1)\)
Recommended article: Q5258114
Recommended article: Q4914710
Recommended article: Q4366334
Recommended article: The global existence of one type of nonlinear Kirchhoff string equation
Recommended article: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings
Recommended article: Global solutions to some nonlinear dissipative mildly degenerate Kirchhoff equations
description: scientific article; zbMATH DE number 5077857