Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates
type: Item
instance of: scholarly article
title: Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates
zbMATH Open document ID: 1162.35008
author: Marina Ghisi
author: Massimo Gobbino
published in: Journal of Differential Equations
publication date: 2008-11-12T00:00:00Z
review text: Equations of Kirchhoff type have been intensively studied in the recent years. Here, decay estimates, as \(t\to\infty\), are obtained for solutions to the abstract setting of the first order problem \[ Bu=u^{\prime}+m(| A^{\frac{1}{2}}u| ^2)Au=0,\quad\quad u(0)=u_o\tag{1} \] and to the singularly perturbed second order problem \[ \epsilon u^{\prime\prime}_{\epsilon}+Bu_{\epsilon}=0,\quad\quad u_{\epsilon}(0)=u_o, \quad\quad u^{\prime}_{\epsilon}(0)=u_1\tag{2} \] where \(A\) is a self-adjoint non negative operator on a Hilbert space \(H\), with dense domain \(D(A)\), \(m:[0,\infty)\to[0,\infty)\) is a \(C^{1}\)-function, and \((u_o,u_1)\in D(A)\times D(A^{\frac{1}{2}}).\) Conditions of nondegenerescence are imposed on \((u_0,u_1)\). Most of the constants which appear in decay estimates relative to problem (2) are independent on \(\epsilon\), if \(\epsilon\) is small enough. Clear tables summarize the comparison with decay estimates stated, previously, in the literature, when \(A\) is coercive or noncoercive.
reviewed by: Denise Huet
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 35L90
zbMATH DE Number: 5365159
zbMATH Keywords: singular perturbations
zbMATH Keywords: degenerate parabolic equations
zbMATH Keywords: degenerate damped hyperbolic equations
zbMATH Keywords: Kirchhoff equations
zbMATH Keywords: decay estimates
MaRDI profile type: Publication
full work available at URL: j.jde.2008.04.017
OpenAlex ID: W4235730673
cites work: A class of explicitly resolvable evolution equations
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 non-Lipschitz nonlinear term
cites work: Q4654968
cites work: Q4239480
cites work: Singular perturbation hyperbolic-parabolic for degenerate nonlinear equations of Kirchhoff type
cites work: Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type
cites work: Global solvability for Kirchhoff equation in special classes of non-analytic functions
cites work: Q4000869
cites work: On the global solvability of Kirchhoff equation for non-analytic initial data
cites work: An evolution equation with non-local non-linearities: Existence, uniqueness and asymptotic behaviour
cites work: Decay properties of solutions to degenerate wave equations with dissipative terms
cites work: Time decay of solutions to degenerate Kirchhoff type equation
cites work: Exponential decay of solutions of some quasilinear hyperbolic equations with linear damping
cites work: Q3480564
cites work: Q3202567
cites work: SHARP DECAY ESTIMATES OF SOLUTIONS FOR MILDLY DEGENERATE DISSIPATIVE KIRCHHOFF EQUATIONS
cites work: Q4269423
cites work: Q4253589
cites work: Q4955159
cites work: The Cauchy problem for Kirchhoff equations
cites work: On some quasilinear wave equations with dissipative terms
DOI: 10.1016/J.JDE.2008.04.017
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description: scientific article; zbMATH DE number 5365159