Viscoelastic Wave Equation Research Articles

Exponential decay and blow-up results for a viscoelastic equation with variable sources

This work deals with a class of nonlinear viscoelastic wave equations with strong damping and variable exponent sources. The main interest of this paper compared to many previous works in the literature is able to use the idea of potential well method to study both the finite time blow-up for solutions starting from the unstable sets and decay estimate for global solutions starting in the potential wells.

Study of a Viscoelastic Wave Equation with a Strong Damping and Variable Exponents

Study of a Viscoelastic Wave Equation with a Strong Damping and Variable Exponents

Energy decay for a viscoelastic wave equation with space-time potential in [formula omitted

In this paper, we consider the following viscoelastic wave equation{utt−(Δu−∫0tg(t−s)Δu(s)ds)+b(t,x)ut+|u|p−1u=0,t>0,x∈Rnu(0,x)=u0(x),ut(0,x)=u1(x),x∈Rn with space-time dependent potential and where the initial data u0(x), u1(x) have compact support. Under suitable assumptions on the relaxation function g and the potential b, we obtain a more general energy decay result using the perturbed energy method.

General energy decay and exponential instability to a nonlinear dissipative-dispersive viscoelastic Petrovsky equation

This work is concerned with the initial boundary valueproblem for a nonlinear viscoelastic Petrovsky wave equation$$u_{tt}+\Delta^{2}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau-\Delta u_{t}-\Delta u_{tt}+u_{t}|u_{t}|^{m-1}=u|u|^{p-1}.$$ Under suitable conditions on the relaxation function $g$, the globalexistence of solutions is obtained without any relation between$m$ and $p$. The uniform decay of solutions is proved by adaptingthe perturbed energy method. For $p>m$ and sufficient conditionson $g$, an unboundedness result of solutions is also obtained.

A blow-up result for a system of coupled viscoelastic equations with arbitrary positive initial energy

This article is devoted to a study of the blow-up result for a system of coupled viscoelastic wave equations. By establishing a new auxiliary function and using the reduction to absurdity method, we obtain some sufficient conditions on initial data such that the solution blows up in finite time at arbitrarily high initial energy. This work generalizes and improves earlier results in the literature.

Existence and decay of solutions for a higher-order viscoelastic wave equation with logarithmic nonlinearity

The main goal of this paper is to study for the local existence and decay estimates results for a high-order viscoelastic wave equation with logarithmic nonlinerity. We obtain several results: Firstly, by using Feado-Galerkin method and a logaritmic Sobolev inequality, we proved local existence of solutions. Later, we proved general decay results of solutions.

High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via an N -order iterative scheme is established, and then the N -order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N = 2 , a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

A Highly Efficient Reduced-Order Extrapolating Model for the 2D Viscoelastic Wave Equation

A Highly Efficient Reduced-Order Extrapolating Model for the 2D Viscoelastic Wave Equation

The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity

<p style='text-indent:20px;'>This paper deals with the following viscoelastic wave equation with a strong damping and logarithmic nonlinearity:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}-\Delta u+\int_0^tg(t-s)\Delta u(s)ds-\Delta u_t = |u|^{p-2}u\ln|u|. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>A finite time blow-up result is proved for high initial energy. Meanwhile, the lifespan of the weak solution is discussed. The present results in this paper complement and improve the previous work that is obtained by Ha and Park [<i>Adv. Differ. Equ.</i>, (2020) 2020: 235].</p>

Blow-Up for a Stochastic Viscoelastic Lamé Equation with Logarithmic Nonlinearity

In this paper, we consider an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and logarithmic nonlinear source terms. We proved a blow-up result for the solution with decreasing kernel.

Wave Propagation in a Fractional Viscoelastic Tissue Model: Application to Transluminal Procedures.

In this article, a wave propagation model is presented as the first step in the development of a new type of transluminal procedure for performing elastography. Elastography is a medical imaging modality for mapping the elastic properties of soft tissue. The wave propagation model is based on a Kelvin Voigt Fractional Derivative (KVFD) viscoelastic wave equation, and is numerically solved using a Finite Difference Time Domain (FDTD) method. Fractional rheological models, such as the KVFD, are particularly well suited to model the viscoelastic response of soft tissue in elastography. The transluminal procedure is based on the transmission and detection of shear waves through the luminal wall. Shear waves travelling through the tissue are perturbed after encountering areas of altered elasticity. These perturbations carry information of medical interest that can be extracted by solving the inverse problem. Scattering from prostate tumours is used as an example application to test the model. In silico results demonstrate that shear waves are satisfactorily transmitted through the luminal wall and that echoes, coming from reflected energy at the edges of an area of altered elasticity, which are feasibly detectable by using the transluminal approach. The model here presented provides a useful tool to establish the feasibility of transluminal procedures based on wave propagation and its interaction with the mechanical properties of the tissue outside the lumen.

Local existence and blow up of solutions for a logarithmic nonlinear viscoelastic wave equation with delay

In this work, we consider a logarithmic nonlinear viscoelastic wave equation with a delay term in a bounded domain. We obtain the local existence of the solution by using the Faedo-Galerkin approximation. Then, under suitable conditions, we prove the blow up of solutions in finite time.

Estimates for the linear viscoelastic damped wave equation on the Heisenberg group

In the paper, we investigate the linear wave equation with viscoelastic damping on the Heisenberg group Hn. Basing on the group Fourier transform on Hn and on the properties of the Hermite functions, we derive some L2(Hn)−L2(Hn) estimates with additional L1(Hn) regularity on initial data for the solution and its higher-order horizontal gradients.

An all‐at‐once approach to full waveform inversion in the viscoelastic regime

Full waveform seismic inversion (FWI) in the viscoelastic regime entails the task of identifying parameters in the viscoelastic wave equation from partial waveform measurements. Traditionally, one frames this nonlinear problem as an operator equation for the parameter‐to‐state map. Alternatively, in an all‐at‐once approach, one augments the nonlinear operator by the viscoelastic wave equation as an additional component and considers the states as additional variables. Hence, parameters and states are sought for simultaneously. In this article, we give a mathematically rigorous all‐at‐once version of FWI in a functional analytical formulation. Further, the corresponding nonlinear map is shown to be Fréchet differentiable, and the adjoint operator of the Fréchet derivative is given in an explicit way suitable for implementation in a Newton‐type/gradient‐based regularization scheme.

Correlative least-squares reverse time migration in viscoelastic media

Standard elastic least-squares reverse time migration (LSRTM) generally ignores the subsurface attenuation effects which can lead to blurred and unfocused migration images. To correct these attenuation effects, we develop a correlative Q-LSRTM approach in viscoelastic media. We use the viscoelastic wave equation based on the standard linear solid model to simulate the subsurface attenuation. The Born modeling and adjoint state equations in viscoelastic media can be derived by the Born approximation and adjoint state method, respectively. Instead of the conventional amplitude-matching misfit function, the proposed Q-LSRTM approach utilizes the normalized zero-lag crosscorrelation misfit function to emphasize the phase measure between the predicted and observed data. The gradients of the crosscorrelation misfit function with regard to reflectivity images of P- and S-wave impedance are estimated by a zero-lag crosscorrelation between the forward background wavefields and the reversal adjoint wavefields. In view of computational efficiency, we also derive an analytical step-length formula for the correlative Q-LSRTM approach, which only requires to solve the Born modeling operator once. Numerical examples using several 2D models have determined that the proposed Q-LSRTM can improve better image quality, compared with the standard elastic LSRTM approach. Furthermore, the crosscorrelation misfit function is less sensitive to amplitude errors.

General decay results for a viscoelastic wave equation with a variable exponent nonlinearity

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

Prestack waveform inversion based on analytical solution of the viscoelastic wave equation

Amplitude-variation-with-offset (AVO) inversion is based on single interface reflectivity equations. It involves some restrictions, such as the small-angle approximation, including only primary reflections, and ignoring attenuation. To address these shortcomings, the analytical solution of the 1D viscoelastic wave equation is used as the forward modeling engine for prestack inversion. This method can conveniently handle the attenuation and generate the full wavefield response of a layered medium. To avoid numerical difficulties in the analytical solution, the compound matrix method is applied to rapidly obtain the analytical solution by loop vectorization. Unlike full-waveform inversion, the proposed prestack waveform inversion (PWI) can be performed in a target-oriented way and can be applied in reservoir study. Assuming that a Q value is known, PWI is applied to synthetic data to estimate elastic parameters including compressional wave (P-wave) and shear wave (S-wave) velocities and density. After validating our method on synthetic data, this method is applied to a reservoir characterization case study. The results indicate that the reflectivity calculated by our approach is more realistic than that computed by using single interface reflectivity equations. Attenuation is an integral effect on seismic reflection; therefore, the sensitivity of seismic reflection to P-and S-wave velocities and density is significantly greater than that to Q, and the seismic records are sensitive to the low-frequency trend of Q. Thus, we can invert for the three elastic parameters by applying the fixed low-frequency trend of Q. In terms of resolution and accuracy of synthetic and real inversion results, our approach performs superiorly compared to AVO inversion.

Nonexistence of global solutions for a class of viscoelastic wave equations

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

Exponential decay for the semilinear wave equation with localized frictional and Kelvin–Voigt dissipating mechanisms

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

Existence and decay of solution to coupled system of viscoelastic wave equations with strong damping in Rn

In this paper, we establish a general decay rate properties of solutions for a coupled system of viscoelastic wave equations in IRn under some assumptions on g1; g2 and linear forcing terms. We exploit a density function to introduce weighted spaces for solutions and using an appropriate perturbed energy method. The questions of global existence in the nonlinear cases is also proved in Sobolev spaces using the well known Galerkin method.