Local energy decay for linear wave equations with non‐compactly supported initial data

A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14 :561–568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata–Matsuyama (Sci. Math. Japon. 2002; 55 :33–42) and Todorova–Yordanov (J. Differential Equations 2001; 174 :464). Copyright © 2004 John Wiley & Sons, Ltd.     

Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)_utt−div(A(x)∇u)+a(x)_ut=0$c(x)u_{tt}-\mathrm{div}(A(x)\mathrm{\nabla }u)+a(x)u_t=0$ in Rⁿ${\mathbb{R}}^n$ are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L²$L^2$ estimates for spatial derivatives.     

Local energy decay for a class of hyperbolic equations with constant coefficients near infinity

A uniform local energy decay result is derived to a compactly perturbed hyperbolic equation with spatial vari¬able coefficients. We shall deal with this equation in an N ‐dimensional exterior domain with a star‐shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data and the equation includes anisotropic variable coefficients {a_i(x): i = 1, 2, …, N }, which are not necessarily equal to each other (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

Oscillation criteria for first-order neutral nonlinear difference equations with variable coefficients

Consider the first-order neutral nonlinear difference equation of the form

, where τ > 0, σ_i ≥ 0 (i = 1, 2,…, m) are integers, {p_n} and {q_n} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ_n=0^∞ q_n = ∞ or Σ_n=0^∞ nq_n Σ_j=n^∞ q_j = ∞ commonly used in the literature.     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.     

Exponential decay domain of energy for wave equation under feedback control

1.IntroductionLetnbeaboundedopendomaininH"(n22)withsmoothboundaryr,u~u(s)betheouternormalofrat8.GivenavectorxoEE",denotem(x)=x--xo(xER"),R=sup{im(x)IIxEfi}.SetWeconsiderthefollowingwaveequstiononnwithboundaryfeedback:whereK=(m'u)k(8),L=f(m'u),with(,n...     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

Local energy decay in linear elastodynamics

We prove that the solutions of a general initial boundary‐value problem of linear elastodynamics in an unbounded domain tend locally to zero as the time tends to +∞, provided the acoustic tensor satisfies the hyperbolicity condition and a non‐trapping hypothesis is assumed. Copyright © 2010 John Wiley & Sons, Ltd.     

Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions

A variable coefficient viscoelastic wave equation with acoustic boundary conditions and nonlinear source term is considered. Under suitable conditions on the initial data and the relaxation function g, we show the polynomial decay of the energy solution and the blow up of solutions by energy methods. The estimates for the lifespan of solutions are also given.     

Uniform energy decay of Schrodinger equations with variable coefficients

This paper establishes the exponential decay of the solutionsof the Schrödinger equation with variable coefficientsin the principal part subject to dissipative feedback actingin the Dirichlet boundary condition. The approach adopted usesa lifting argument in the topology of the solution along withRiemannian geometry method.     

Exact controllability of the wave equation with time-dependent and variable coefficients

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with variable coefficients in time and space, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric conditions.     

Exponential energy decay of solutions of viscoelastic wave equations

In this paper we consider the nonlinear viscoelastic equation

     

Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain

Uniform energy and L² decay of solutions for linear wave equations with localized dissipation will be given. In order to derive the L²-decay property of the solution, a useful device whose idea comes from Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) is used. In fact, we shall show that the L²-norm and the total energy of solutions, respectively, decay like and O(1/t²) as t→+∞ for a kind of the weighted initial data.     

具时间依赖系数和耗散边界的非线性波方程的能量指数衰减性

利用能量法证明了具耗散边界条件和时间依赖系数的非线性波方程的能量指数衰减性.     

Local energy decay for the wave equation with a nonlinear time-dependent damping

This article addresses a wave equation on a exterior domain in ? ^d (d odd) with nonlinear time-dependent dissipation. Under a microlocal geometric condition we prove that the decay rates of the local energy functional are obtained by solving a nonlinear non-autonomous differential equation     

Generalised energy conservation law for wave equations with variable propagation speed

We investigate the long time behaviour of the L²-energy of solutions to wave equations with variable speed of propagation. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.     

The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source

The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form