Problem For Wave Equation Research Articles

Existence of Global Decaying Solutions to the Cauchy Problem for a Nonlinear Dissipative Wave Equation of Klein-Gordon Type with a Derivative Nonlinearity

We prove the existence of global decaying solutions to the Cauchy problem for the wave equation of Klein-Gordon type with a nonlinear dissipation and a derivative nonlinearity. To derive required estimates of solutions we employ a delicate 'loan' method.

Spectral Approach to Derive the Representation Formulae for Solutions of the Wave Equation

Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well‐known d′Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.

Well- and ill-posedness issues for energy supercritical waves

We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. The proof uses the finite speed of propagation and a quantitative study of the associated ODE. It does not require any scaling invariance of the equation. We also obtain some ill-posedness and weak ill-posedness results.

Convergence estimate of the RBF-based meshless method for initial-boundary value problem of wave equations

This paper gives an order of convergence in applying the radial basis functions (RBF) as a meshless method for the initial-boundary value problem of wave equations. The analysis for a model wave equation shows that the convergence order of the proposed method is at least the same as which in finite difference method (FDM). For illustration, two examples are given to verify the accuracy and efficiency of the numerical solution by means of RBF and FDM.

An identification problem for a time periodic nonlinear wave equation in non cylindrical domains

The existence of a time-periodic solution of a free boundary nonlinear wave equation in non cylindrical domains is established. The problem arises in the study of the identification of the coefficient of the wave equation and of the boundary of the region from the observed values of the solution in a fixed subregion.

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations u t t − u x x = b ( u , D u ) u x x + 2 a 0 ( u , D u ) u t x + F ( u , D u ) on exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations.

Optimal control problems for wave equations

The problem on minimizing a quadratic functional on trajectories of the wave equation is considered. We assume that the density of external forces is a control function. A control problem for a partial differential equation is reduced to a control problem for a countable system of ordinary differential equations by use of the Fourier method. The controllability problem for this countable system is considered. Conditions of the noncontrollability for some wave equations were obtained.

A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

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.

Wave equations with super-critical interior and boundary nonlinearities

AbstractThis article presents a unified overview of the latest, to date, results on boundary value problems for wave equations with super-critical nonlinear sources on both the interior and the boundary of a bounded domain Ω∈Rn. The presented theorems include Hadamard local wellposedness, global existence, blow-up and non-existence theorems, as well as estimates on the uniform energy dissipation rates for the appropriate classes of solutions.

Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher-order wave equations

The Keel-Smith-Sogge estimates are considered in a general frame in terms of the harmonic analysis. The estimates are applied to the Cauchy problem for nonlinear higher-order wave equations, and the existence of global and almost-global solutions is shown.

Two-point boundary value problems and exact controllability for several kinds of linear and nonlinear wave equations

All articles must In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.

Uniform decay rate for the transmission wave equations with variable coefficients

This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boundary, the author establishes the uniform energy decay rate for the problem.

Finite time blow up of solutions for wave equations with two non-linear source terms of different sign in critical energy case

We study the initial boundary value problem of wave equations with two nonlinear source terms of different sign in the critical energy case. By the standard convex method, it is proven that the solution of the problem does not exist globally in time. And it is pointed out that the solution blows up in finite time under some assumption regarding the initial data. Key word: Wave equation, finite time blow up, initial boundary problem.

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R n . In this paper, we study half space problems in R + n = R + × R n − 1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x ′ ∈ R n − 1 . For the variable x 1 ∈ R + in the normal direction, we use L 2 space or weighted L 2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ . The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].

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.

Simulation of low-frequency sound propagation in an irregular shallow-water waveguide with a fluid bottom

A further development of a previously proposed approach to calculating the sound field in an arbitrarily irregular ocean is presented. The approach is based on solving the first-order causal mode equations, which are equivalent to the boundary-value problem for acoustic wave equations in terms of the cross-section method. For the mode functions depending on the horizontal coordinate, additional terms are introduced in the cross-section equations to allow for the multilayer structure of the medium. A numerical solution to the causal equations is sought using the fundamental matrix equation. For the modes of the discrete spectrum and two fixed low frequencies, calculations are performed for an irregular two-layer waveguide model with fluid sediments, which is close to the actual conditions of low-frequency sound propagation in the coastal zone of the oceanic shelf. The calculated propagation loss curves are used as an example for comparison with results that can be obtained for the given waveguide model with the use of adiabatic and one-way propagation approximations.

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial–boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial–boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.

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

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

Cauchy problem for semilinear wave equation with time-dependent metrics

We establish the existence of a weak solution u of the semilinear wave equation ∂ t 2 u − d i v x ( a ( t , x ) ∇ x u ) = f k ( u ) where a ( t , x ) is equal to 1 outside a compact set with respect to x and a non-linear term f k which satisfies | f k ( u ) | ≤ C | u | k . For some non-trapping time-periodic perturbations a ( t , x ) , we obtain the long time existence of a solution from little initial data.