Value Problem Of Wave Equation Research Articles

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime

The main goal of this work is to discuss formation of singularities of solutions to the Cauchy problems for single wave equation and coupled system of wave equations with nonlinear memory terms in Schwarzschild spacetime. For the initial value problem of nonlinear wave equations, blow-up results of sign changing solutions are derived by employing the suitable test function technique which is related to the Riemann-Liouville fractional derivative and contradiction argument. Additionally, upper bound lifespan estimates of solutions to the small initial value problem of wave equations are established by imposing certain conditions on the initial values. It is worth to mention that crucial difficulty is the setting of test function framework, which has not been applied to the Schwarzschild spacetime in the previous literatures. To the best of our knowledge, the results in Theorems 1.1-1.4 are new.

The flexibility in choosing distinct Green’s functions for the boundary wall method: waveguides and billiards

The boundary wall method (BWM) is a general purpose protocol to treat boundary value problems for wave equations, specially Helmholtz’s (the case addressed here). Similarly to most approaches, the BWM may be computationally demanding for large borders , at which the wave function must satisfy specified boundary conditions. Also, despite the fact the BWM is an exact procedure, usually it is not amenable to closed form solutions. The BWM relies on the Green’s function G 0 of the embedding domain V of . However, in many instances—like for modeling a billiard—the specific V is not really fundamental and thus one has a certain freedom to choose distinct domains and so G 0’s. Here we consider this characteristic of the BWM and show how to obtain some analytical results and solve numerically semi-infinite waveguides by exploring proper Green’s functions. As examples, we discuss rectangular, triangular and trapezoidal structures with both Dirichlet and leaking boundaries as well as scattering states within semi-infinite rectangular waveguides.

Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.

Lorentzian Calderón problem under curvature bounds

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior region of double null cones. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.

Inverse boundary value problems for wave equations with quadratic nonlinearities

We study inverse problems for the nonlinear wave equation □gu+w(x,u,∇gu)=0 in a Lorentzian manifold (M,g) with boundary, where ∇gu denotes the gradient and w(x,u,ξ) is smooth and quadratic in ξ. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric g can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of w(x,u,ξ) with respect to u up to null forms.

An integral evolution formula of boundary value problem for wave equations

This article proposes a new integral evolution formula of boundary value problem for wave equations of the form utt(x,t)+L(x,D)u(x,t)=f(x,t). By introducing the operator functions, e.g., ϕ-functions, and using the Duhamel’s principle, a compact integral evolution formula is established for inhomogeneous wave equations. The derivation is based on Duhamel’s principle and the theory of operational calculus.

On solving wave equations on fixed bounded intervals involving Robin boundary conditions with time-dependent coefficients

In this paper, it is shown how characteristic coordinates, or equivalently how the well-known formula of d’Alembert, can be used to solve initial-boundary value problems for wave equations on fixed, bounded intervals involving Robin type of boundary conditions with time-dependent coefficients. A Robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space-derivative on a boundary of the interval. Analytical methods, such as the method of separation of variables (SOV) or the Laplace transform method, are not applicable to those types of problems. The obtained analytical results by applying the proposed method, are in complete agreement with those obtained by using the numerical, finite difference method. For problems with time-independent coefficients in the Robin boundary condition(s), the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables, or by the finite difference method.

Blow up for initial-boundary value problem of wave equation with a nonlinear memory in 1-D

The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: $${u_{tt}} - {u_{xx}} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int_0^t {{{\left( {t - s} \right)}^{ - \gamma }}{{\left| {u\left( s \right)} \right|}^p}ds} .$$ The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.

Initial Value Problems for Wave Equations on Manifolds

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hörmander.

A Fictitious Time Integration Method For A One-dimensional Hyperbolic Boundary Value Problem

This paper gives an estimate for the initial-boundary value problem of wave equations by using the Fictitious Time Integration Method (FTIM) previously developed by Liu and Atluri [1]. Given examples confirm that FTIM is highly efficient approach to find the true solutions. It is interesting that the FTIM can easily treat the boundary value problems without any iteration and has high efficiency and high accuracy.

Conformally Covariant Systems of Wave Equations and their Equivalence to Einstein’s Field Equations

We derive, in 3 + 1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich’s conformal field equations. We analyse their equivalence to Einstein’s vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.

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.

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.

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.

Some generalized results for global well-poseness for wave equations with damping and source terms

In this paper we study the initial boundary value problem of wave equations with nonlinear damping and source terms: u t t − Δ u + a | u t | m − 1 u t = b | u | p − 1 u , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( x , t ) = 0 , x ∈ ∂ Ω , t ≥ 0 , where Ω ⊂ R N is a suitably smooth bounded domain. We prove that for any a > 0 and b > 0 , if 1 < p < m < ∞ , u 0 ( x ) ∈ H 0 1 ( Ω ) ∩ L p + 1 ( Ω ) , u 1 ( x ) ∈ L 2 ( Ω ) , then for any T > 0 , above problem admits a global solution u ( x , t ) ∈ L ∞ ( 0 , T ; H 0 1 ( Ω ) ∩ L p + 1 ( Ω ) ) with u t ( x , t ) ∈ L ∞ ( 0 , T ; L 2 ( Ω ) ) ∩ L m + 1 ( Ω × [ 0 , T ] ) . So the results of Georgiev and Ikehata are generalized and improved.

Remarks on wave equations involving two opposite nonlinear source terms

We study the initial boundary value problem of wave equations involving two opposite nonlinear source terms. By standard convex argument, 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.

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invariance of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.

Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign

We study the initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms of different sign. By introducing a family of potential wells $W_\delta$ and corresponding outside sets $V_\delta$ of $W_\delta$ we first obtain the invariant sets and vacuum isolating of solutions. Then we get the threshold result of global existence and nonexistence of solutions. Finally we prove the global existence of solutions for the problem with critical initial conditions $I(u_0)\ge 0$, $E(0)=d$ (or $J(u_0)=d$).

Exact and approximate artificial boundary conditions for the hyperbolic problems in unbounded domains

In this paper, the numerical solutions of the problems of wave equation in two dimensions in unbounded domains are considered. For a given problem, we introduce an artificial boundary B to make the computational domain finite. Three kinds of equivalent exact and approximate artificial boundary conditions are obtained on circular artificial boundary B by a constructive method. On the artificial boundary B , we propose an exact or approximate boundary condition to reduce the given problem an initial-boundary value problem of wave equation in the finite computational domain, which is equivalent to the original problem. Then the finite difference method and finite element method are used to solve the reduced problem in the finite computational domain. Some numerical examples are presented to show the feasibility and effectiveness of the method given in this paper.