Complete second order differential equations in Banach spaces with dynamic boundary conditions
type: Item
instance of: scholarly article
title: Complete second order differential equations in Banach spaces with dynamic boundary conditions
zbMATH Open document ID: 1059.34034
author: Jin Liang
author: Ti-Jun Xiao
published in: Journal of Differential Equations
publication date: 2004-10-01T00:00:00Z
review text: The authors consider the following second-order problem with dynamic boundary conditions in a Banach space \[ \begin{gathered} u''(t)+ Au(t)+ Bu'(t)= 0,\quad t\geq 0,\\ x''(t)+ A_1x(t)+ B_1x'(t)= G_0u(t)+ G_1u'(t),\quad t\geq 0,\\ x(t)= Pu(t),\quad t\geq 0,\\ u(0)= u_0,\;x(0)= x_0, u'(0)= u_1,\;x'(0)= x_1.\end{gathered} \] \(A\), \(B\), \(A_1\), \(B_1\), \(G_0\), \(G_1\) and \(P\) are linear operators. The second equation listed above acts as the dynamic boundary condition for \(u\). The authors overview a number of results for first-order problems with dynamic boundary conditions, but note that there has been less work on the second-order case. They also mention that while this problem can be converted into a first-order problem, this necessitates adding further assumptions. They deal with the second-order problem above by converting it into an associated abstract second-order Cauchy problem with operator matrices as coefficients. In Section 2, after recalling the definition of strong wellposedness for such problems, they introduce the notion of strong quasi-wellposedness. This definition is motivated by the fact that it is easier to verify it and is more general than strong wellposedness. They also specify conditions under which the two definitions are equivalent. In Section 3, they prove Theorem 3.2, which specifies necessary and sufficient conditions for the strong wellposedness of the associated Cauchy problem and Theorem 3.5, which gives sufficient conditions for strong quasi-wellposedness. Section 4 considers the inhomogeneous case. In Section 5, the authors give a number of initial boundary value problems involving partial differential equations to which these results can be applied. A lot of detail is given in these examples, which is of great convenience to the reader.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 35G05
zbMATH DE Number: 2104507
zbMATH Keywords: Second-order differential equations in Banach spaces
zbMATH Keywords: Mixed initial boundary value problem
zbMATH Keywords: Dynamic boundary conditions
zbMATH Keywords: wellposedness
zbMATH Keywords: inhomogeneous
zbMATH Keywords: Hyperbolic equations
zbMATH Keywords: Wentzell boundary conditions
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
full work available at URL: j.jde.2004.01.011
OpenAlex ID: W1980665242
cites work: Second order evolution equations with dynamic boundary conditions
cites work: A semigroup approach to boundary feedback systems
cites work: Quasilinear parabolic systems with dynamical boundary conditions
cites work: Singular perturbation and boundary layer for an abstract Cauchy problem
cites work: Q3678037
cites work: $C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions
cites work: Q4821158
cites work: The heat equation with generalized Wentzell boundary condition.
cites work: Conditions for parabolicity of second order abstract differential equations
cites work: Some nonlinear wave equations with acoustic boundary conditions
cites work: Oscillatory boundary conditions for acoustic wave equations
cites work: Q3781625
cites work: Dynamic boundary conditions for the Navier–Stokes equations
cites work: Evolution equations with dynamic boundary conditions
cites work: Decay of solutions of wave equations in a bounded region with boundary dissipation
cites work: Q3278561
cites work: Q5593959
cites work: Q4767635
cites work: Q4169288
cites work: Q3810985
cites work: On complete second order linear differential equations in Banach spaces
cites work: Differential operators and \(C\)-wellposedness of complete second order abstract Cauchy problems
cites work: The Cauchy problem for higher-order abstract differential equations
cites work: A solution to an open problem for wave equations with generalized Wentzell boundary conditions
DOI: 10.1016/J.JDE.2004.01.011
description: scientific article; zbMATH DE number 2104507