Cauchy Problem For Wave Equations Research Articles

Fast energy decay for wave equation with a monotone potential and an effective damping

We consider the total energy decay of the Cauchy problem for wave equations with a potential and an effective damping. We treat it in the whole one-dimensional Euclidean space [Formula: see text]. Fast energy decay like [Formula: see text] is established with the help of potential. The proofs of main results rely on a multiplier method and modified techniques adopted in [R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping, Nonlinear Anal. 69(4) (2008) 1396–1401].

On Stationary Nonequilibrium Measures for Wave Equations

In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite mean energy density and study the convergence of distributions of the solutions to a limiting Gaussian measure for large times. We derive the formulas for the limiting energy current density (in mean) and find a new class of stationary nonequilibrium states for the studied model.

Uniform Energy Decay for Wave Equations with Unbounded Damping Coefficients

We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.

A note on decay rates of the local energy for wave equations with Lipschitz wavespeeds

We consider the Cauchy problem for wave equations with variable coefficients in the whole space Rn. We improve the rate of decay of the local energy, which has been recently studied by J. Shapiro [11], where he derives the log-order decay rates of the local energy under stronger assumptions on the regularity of the initial data.

On second order weakly hyperbolic equations with oscillating coefficients

We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form $\partial_t^2 u -a(t) \partial_x^2 u=0$. In the strictly hyperbolic case $a (t)\geq c(> 0)$, Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume H$\ddot { \rm o}$lder regularity of the coefficients in order to prove Gevrey well-posedness. Recently, assumptions bearing on the oscillations of the coefficient have been imposed in the literature in order to guarantee well-posedness. In the weakly hyperbolic case $a(t) \geq 0$, Colombini, Jannelli and Spagnolo proved well-posedness in Gevrey classes of order $1\leq s<s_0=1+(k+\alpha)/2$. In this paper, we put forward condition $ |a'(t)| \leq Ca(t)^p/t^q $ that bears both on the oscillations and the degree of degeneracy of the coefficient. we show that under such a condition, Gevrey well-posedness holds for $1\leq s<qs_0/\{ (k+\alpha) (1-p) +q-1\}$ if $q \geq 3-2p$. In particular, this improves on the result (corresponding to the case $p=0$) of Colombini, Del Santo and Kinoshita.

Finite time blow up for a wave equation with nonlinear memory in Schwarzschild spacetime

In this work, we study the Cauchy problem of wave equation with nonlinear memory in Schwarzschild spacetime. Blow up result will be established when the power p of the memory and the data satisfy some conditions. For this purpose, we use two test functions to construct a functional. After that by introducing an iteration argument we conclude that the functional satisfies the conditions of Kato type lemma.

Global Cauchy problem for wave equations with a nonlinear damping term

We study the global Cauchy problem for wave equations with a nonlinear damping term. The existence, uniqueness and nonuniqueness, and nonexistence of a global classical solution are considered.

The critical nonlinear wave equation in two space dimensions

Extending our previous work, we show that the Cauchy problem for wave equations with critical exponential nonlinearities in 2 space dimensions is globally well-posed for arbitrary smooth initial data.

On the existence and uniqueness of solutions of the Cauchy problem for wave equations with general stochastic measures

On the existence and uniqueness of solutions of the Cauchy problem for wave equations with general stochastic measures

On the asymptotic boundedness of the energy of solutions of the wave equation u tt − a(t)Δu = 0

Applying a class of quadratic forms introduced by Manfrin (Port. Math. 65(4):447–484, 2008), we study the asymptotic behavior, as t → +∞, of the energy of the solutions to the Cauchy problem for wave equations with time dependent propagation speed.

A ‘Super-Critical’ Nonlinear Wave Equation in 2 Space Dimensions

These notes are an extended exposion of lectures given at the conference “Nonlinear Analysis”, Verbania, Sept. 25–29, 2010, where we reviewed the results from [11] on global well-posedness of the Cauchy problem for wave equations with exponential nonlinearities in 2 space dimensions for smooth, arbitrarily large radially symmetric data.

Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions

Extending the work of Ibrahim et al. (Commun Pure Appl Math 59(11): 1639–1658, 2006) on the Cauchy problem for wave equations with exponential nonlinearities in two space dimensions, we establish global well-posedness also in the super-critical regime of large energies for smooth, radially symmetric data.

Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.

The Cauchy problem for wave equations with non Lipschitz coefficients; Application to continuation of solutions of some nonlinear wave equations

On considere le probleme de Cauchy pour des equations d'onde strictement hyperboliques : Lu := Σ n j,k=0 ∂yj (α j,,k ∂ yk u) + Σ n j =0{b j ∂ yj u + ∂ yj , (cju)} + du = f, quand les coefficients de la partie principale sont seulement Log-Lipschitz en toutes les variables. Cette classe d'equation est invariante par changement de variables et est donc une classe naturelle pour une etude locale intrinseque. En particulier, on montre l'existence locale, l'unicite locale et la vitesse finie de propagation pour le probleme de Cauchy non caracteristique, donnant une version invariante d'un resultat anterieur du premier auteur avec N. Lerner [6]. Pour les equations non lineaires ou les coefficients ci-dessus dependent de u, la methode d'estimations permet de montrer que les solutions regulieres se prolongent en solutions regulieres aussi longtemps qu'elles restent Log-Lipschitz.

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples.

Wave equations with time-dependent dissipation II. Effective dissipation

This article is intended to present a construction of structural representations of solutions to the Cauchy problem for wave equations with time-dependent dissipation above scaling. These representations are used to give estimates of the solution and its derivatives based on L q ( R n ) , q ⩾ 2 . The article represents the second part within a series. In [Jens Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations 222 (2) (2006) 487–514] weak dissipations below scaling were discussed.

Algorithm and mechanization to Cauchy problem of wave equation with linear damping and source terms

This paper discusses the Cauchy problem of wave equation with linear damping and source terms. The author reduces the problem to a parabolic one and uses the corresponding initial value to construct the solutions of the resulting parabolic problem. Thereafter, he obtains a brief algorithmization formula to this problem with mechanization in Maple.

The Poisson Formula Revisited

This note presents a series solution to the Cauchy problem of wave equations which permits the Cauchy problem in space to be introduced in an introductory partial differential equation (PDE) course.

Ergodic Properties of Hyperbolic Equationswith Mixing

The paper proves the ergodicity of the phase flow of the Cauchy problem for wave equations with respect to the limit measure for statistical solutions of this problem under a mixing condition for the initial measure.

Global solutions of Yang-Mills equations on anti-de Sitter spacetime

Anti-de Sitter spacetime is a Cinfinity manifold diffeomorphic to R4, endowed with a Cinfinity metric of hyperbolic signature. However this spacetime is not globally hyperbolic, and the known results about the solution of the Cauchy problem for wave equations on Lorentzian manifolds do not apply, even for a small interval of time and even for linear equations. The author proves the global existence of a solution of the Cauchy problem for the Yang-Mills-Higgs equations on anti-de Sitter spacetime, under the condition that there is no radiation at timelike infinity, a condition that is explained mathematically.