Problem For Wave Equation Research Articles

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.

Long time existence for two-dimension elastic waves

In this paper, for the Cauchy problem of nonlinear elastic wave equations for two-dimension isotropic, homogeneous and hyperelastic materials with small initial data, we give the lifespan estimates for classical solutions.

Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains

We obtain the weak well-posedness results for the linear strongly damped wave equation and the nonlinear Westervelt equation on arbitrary three-dimensional domains with homogeneous Dirichlet boundary conditions. In R2, we prove the well-posedness in the class of NTA domains or their limit domains, obtained as a limit of sequences of NTA domains, characterized by the same geometrical constants. The nonhomogeneous Dirichlet condition is also treated for Sobolev extension domains of Rn with a d-set boundary n−2<d<n preserving Markov's local inequality. For a converging sequence of domains in the sense of characteristic functions, we establish the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition.

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.

A New Method for Blow-Up to Scale-Invariant Damped Wave Equations with Derivatives and Combined Nonlinear Terms

The Cauchy problems of scale-invariant damped wave equations with derivative nonlinear terms and with combined nonlinear terms are studied. A new method is provided to show that the solutions will blow up in a finite time, if the nonlinear powers satisfy some conditions. The method is based on constructing appropriate test functions, by using the solution of an ordinary differential equation. It may be useful to prove the nonexistence for global solutions for other nonlinear evolution equations.

A Numerical Method for the Variable-Order Time-Fractional Wave Equations Based on the H2N2 Approximation

Aiming at the initial boundary value problem of variable-order time-fractional wave equations in one-dimensional space, a numerical method using second-order central difference in space and H2N2 approximation in time is proposed. A finite difference scheme with second-order accuracy in space and 3 − γ ∗ order accuracy in time is obtained. The stability and convergence of the scheme are further discussed by using the discrete energy analysis method. A numerical example shows the effectiveness of the results.

The Quality Properties of Integral Type Problems for Wave Equations and Applications

In this paper, the integral problem for linear and nonlinear wave equations is studied. The equation involves abstract operator A in Hilbert space H. Here, assuming enough smoothness on the initial data and the operators, the existence, uniqueness, regularity properties of solutions are established. By choosing the space H and A, the regularity properties of solutions of a wide class of wave equations in the field of physics are obtained.

On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

&lt;p style='text-indent:20px;'&gt;In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.&lt;/p&gt;

Integral type Cauchy problem for abstract wave equations and applications

We consider the integral type problem for linear and nonlinear abstract wave equations with operator coefficient in E-valued , spaces. The equation includes an abstract linear operator A that defined in a Banach space E. Here, the existence, uniqueness, uniform estimates of solution and quality properties of integral wave problem are derived. We also obtain the same properties for solutions of mixed integral problem for nonlocal and anisotropic classes, by choosing the space E and operator A.

The regularity properties and blow-up of the solutions for nonlocal general wave equations

In this paper, the Cauchy problem for nonlocal linear and nonlinear wave equations are studied. The equations include the general differential operators. The existence, uniqueness, Lp-regularity properties and blow-up at finite time of solutions of the Cauchy problem is obtained. By choosing differential operators including in equations, the regularity properties of a different type wave equations are studied.

Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms

We consider a transmission problem of Kirchhoff type wave equations with delay and damping terms, subject to a memory condition on one part of the boundary. Under appropriate hypotheses on the relaxation function and the relationship between weights of damping and delay terms, we establish well-posedness of the problem. Using the Faedo-Galerkin approximation technique, and introducing suitable energyand Lyapunov functionals, we obtain estimates for exponential, polynomial, and logarithmic decay. For more information see https://ejde.math.txstate.edu/Volumes/2021/95/abstr.html

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.

On a backward problem for nonlinear time fractional wave equations

In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.

The lifespan of classical solutions of semilinear wave equations with spatial weights and compactly supported data in one space dimension

This paper studies initial value problems for semilinear wave equations with spatial weights in one space dimension. The lifespan estimates of classical solutions for compactly supported data are established in all the cases of polynomial weights. The results are classified into two cases according to the total integral of the initial speed.

Global existence for one-dimensional hyperbolic equation with power type nonlinearity

In this paper, we consider the Cauchy problem for one-dimensional damped wave equations with variable coefficients $a(x)$ having power type nonlinearity of order $p$. We discuss the global existence of solutions for small initial data and investigate the relation between the range of $a(x)$ and the order $p$.

Nonexistence of global solutions for damped abstract wave equations with memory

We consider a class of abstract nonlinear wave equations with memory and linear dissipation. We give sufficient conditions in terms of the nitial data to prove the nonexistence of global solutions. We improve recent results that have studied this problem for viscoelastic wave, Kirchhoff and Petrovsky equations with positive initial energy values.

Lifespan estimates for 2-dimensional semilinear wave equations in asymptotically Euclidean exterior domains

In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if 1<p≤pc(2), the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for 1<p≤2. It is interesting to see that the logarithmic increase of the harmonic function in 2-D has no influence to the estimate of the upper bound of the lifespan for 2<p≤pc(2). One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simply connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.

ON COMPACT 4TH ORDER FINITE-DIFFERENCE SCHEMES FOR THE WAVE EQUATION

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the n-dimensional nonhomogeneous wave equation, n≥ 1. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for n≥ 2. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schro¨dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for n≥ 2. For n = 1 and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Global existence results for semi-linear structurally damped wave equations with nonlinear convection

In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term [Formula: see text], where [Formula: see text] is a constant. As is now well known, the linear principal part brings both the diffusion phenomenon and the regularity loss of solutions. This implies that, for the nonlinear problems, the choice of solution spaces plays an important role to obtain the global solutions with the sharp decay properties in time. Our main purpose in this paper is to prove the global (in time) existence of solutions for the small data and their decay properties for the supercritical nonlinearities.

Solution of Inverse Problems for Wave Equation with a Nonlinear Coefficient

Solution of Inverse Problems for Wave Equation with a Nonlinear Coefficient