Weighted energy decay for 1D wave equation

We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.     

Local energy decay for the damped Klein-Gordon equation in exterior domain

We prove uniform local energy decay for the solution of the dissipative Klein–Gordon equation on an exterior domain under some geometric condition called “exterior geometric control”.     

Global existence and energy decay for a damped quasi-linear wave equation

In this paper we prove the global existence and study decay property of the solutions to the initial boundary value problem for the quasi-linear wave equation with a dissipative term without the smallness of the initial data. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

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     

Local energy decay of solutions to the wave equation for nontrapping metrics

We prove uniform local energy decay estimates of solutions to the wave equation on unbounded Riemannian manifolds with nontrapping metrics. These estimates are derived from the properties of the resolvent at high frequency. Applications to a class of asymptotically Euclidean manifolds as well as to perturbations by non-negative long-range potentials are given.     

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Local existence and uniqueness for a semilinear accretive wave equation

We study local existence and uniqueness in the phase space Hμ×Hμ−1(RN) of the solution of the semilinear wave equation utt−Δu=ut|ut|^p−1 for p>1.     

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially even if m > 2.     

On decay properties of solutions for degenerate strongly damped wave equations of Kirchhoff type

We consider the initial-boundary value problem for the degenerate strongly damped wave equations of Kirchhoff type: . For all t?0, we will give the optimal decay estimate C−1(1+t)^−1/γ?‖A1/2u(t)^‖2?C(1+t)^−1/γ, when either the coefficient ρ is appropriately small or the initial data are appropriately small. And, we will show a decay property of the norm ‖Au(t)^‖2 for t?0.     

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.     

Energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source

This paper deals with the energy decay estimates and infinite blow‐up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source term under null Dirichlet boundary condition. By constructing a new family of potential wells, together with logarithmic Sobolev inequality and perturbation energy technique, we establish sufficient conditions to guarantee the solution exists globally or occurs infinite blow‐up and derive the polynomial or exponential energy decay estimates under some appropriate conditions.     

Long time decay for 2D Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein-Gordon equations. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger's type by the spectral approach. For the proof we modify the approach to make it applicable to relativistic equations.     

Asymptotic profile of solutions for the damped wave equation with a nonlinear convection term

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one‐dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.     

Remarks on the decay of the local energy for semilinear wave equation

In this note, we prove the global well posedness and the local energy decay for semilinear wave equation with small data.     

On uniform decay for transmission problem of Kirchhoff type viscoelastic wave equation

Abstract In this paper we consider the large time behavior of solutions to an n-dimensional transmission problem for two Kirchhoff type viscoelastic wave equations, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is a simple elastic part while the other is a viscoelastic component endowed with a long range memory. We show that the dissipation produced by the viscoelastic part is strong enough to produce exponential or polynomial decay of the solution     

Local energy decay for linear wave equations with variable coefficients

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an 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 its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L²-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].