Scattering for a quasilinear hyperbolic equation of Kirchhoff type
type: Item
instance of: scholarly article
title: Scattering for a quasilinear hyperbolic equation of Kirchhoff type
zbMATH Open document ID: 0895.35067
author: Taeko Yamazaki
published in: Journal of Differential Equations
publication date: 1998-09-20T00:00:00Z
review text: The global solvability and the existence of the scattering operator for the quasilinear hyperbolic Kirchhoff type equation \[ {\partial^2 u\over \partial t^2}= m\bigl( \| \nabla u\|^2 \bigr)^2 \Delta u\quad \text{in} \quad \mathbb{R}^n_x \times \mathbb{R}_t \tag{1} \] \[ u(x,0) =\varphi_0 (x),\quad {\partial u\over \partial t} (x,0)= \psi_0 (x) \tag{2} \] are studied, where \(m\in C^1 ([0,\infty)]\) and \(\inf_{\lambda\geq 0} m(\lambda) =m_0>0\). The wave and scattering operators in a neighborhood of the origin in some suitable function space are constructed, and it is shown the continuity of the scattering operator with respect to a suitable topology. Global solvability is proven using an a priori estimate for the local solutions of (1), (2). The asymptotic behavior of the Cauchy problem is also analysed, and an extension of local solvability is given for the abstract form.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1130533
zbMATH Keywords: global solvability
reviewed by: István Ecsedi
MaRDI profile type: Publication
full work available at URL: jdeq.1997.3372
OpenAlex ID: W2013924529
cites work: Asymptotic behaviour as \(t \to +\infty\) of the solutions of linear hyp equations with coefficients discontinuous in time (on a bounded domain)
cites work: On the mildly degenerate Kirchhoff string
cites work: On the Well-Posedness of the Kirchhoff String
cites work: Q4112938
cites work: Global solvability for the degenerate Kirchhoff equation with real analytic data
cites work: A class of nonlinear hyperbolic problems with global solutions
cites work: Nonlinear perturbations of the kirchhoff equation
cites work: Kirchhoff type equations depending on a small parameter
cites work: Local solutions for a nonlinear degenerate Hyperbolic equation
cites work: Asymptotic behaviour for the Kirchhoff equation
cites work: The initial value problem for a stretched string
cites work: On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms
cites work: A class of second order quasilinear evolution equations
cites work: Q5627250
cites work: Mathematical analysis of the nonlinear model for the vibrations of a string
cites work: Q3797606
cites work: Q3709198
cites work: Abstract non linear wave equations
cites work: On some quasilinear wave equations with dissipative terms
cites work: Some nonlinear degenerate wave equations
cites work: Q3031362
DOI: 10.1006/JDEQ.1997.3372
description: scientific article; zbMATH DE number 1130533