Time decay rates for undamped constant coefficients linear partial differential equations
type: Item
instance of: scholarly article
title: Time decay rates for undamped constant coefficients linear partial differential equations
zbMATH Open document ID: 0846.35015
author: Matei Kelemen
published in: Journal of Differential Equations
publication date: 1996-05-13T00:00:00Z
review text: The Cauchy problem for linear partial differential operators of order \(m\) of the type \[ T(\tau, D)= \sum^p_{j= 0} \tau^{p- j} R_j(D), \] \(\tau= \partial/\partial t\), \(D= (\partial/\partial x_1,\dots, \partial/\partial x_n)\), \(R_j\) being polynomials of degree \(d\leq m- p+ j\), is analyzed. The zeros of the characteristic polynomial \(T(s, i\xi)\) are assumed to be in the complex open left half plane, but not necessarily bounded away from the imaginary axis. It is shown that the solution \(y\) tends to zero uniformly in space, as \(t\to \infty\), and decay rates are described. Smoothness of the initial data \(\tau^j y(0, x)\), \(j= 0,\dots, p- 1\), imply fast decay, in general polynomial. For the proof the Fourier transform and explicit representation formulae for the solution are exploited. An example is presented which also illustrates the relation to feedback stabilization.
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 859259
zbMATH Keywords: polynomial decay
zbMATH Keywords: exponential decay
reviewed by: Reinhard Racke
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0030
OpenAlex ID: W2025244591
DOI: 10.1006/JDEQ.1996.0030
Recommended article: On the decay rate of solutions of linear parabolic equations for infinitely increasing time
Recommended article: The rate for decay of solutions in nonlinear dissipative partial differential equations
Recommended article: Q5316245
Recommended article: On decay rate of solution to degenerating linear parabolic equation
Recommended article: Uniform decay rate for the transmission wave equations with variable coefficients
Recommended article: Decay rates of the solutions of nonlinear dispersive equations
Recommended article: Rate of decay to \(0\) of the solutions to a nonlinear parabolic equation
Recommended article: Uniform decay rates for second-order hyperbolic equations with variable coefficients
Recommended article: Time decay for hyperbolic equations with homogeneous symbols
Recommended article: Q4525528
description: scientific article; zbMATH DE number 859259