Viscoelastic Wave Equation Research Articles

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.

General Decay Of A Nonlinear Viscoelastic Wave: Equation With Boundary Dissipation

In this work we establish a general decay rate for a nonlinear viscoelastic wave equation with boundary dissipation where the relaxation function satisfies $$$g^{\prime }\left( t\right) \leq -\xi \left( t\right) g^{p} % \left( t\right) , t\geq 0, 1\leq p\leq \frac{3}{2}.$$$ This work generalizes and improves earlier results in the literature.

Fully Scalable Solver for Frequency-Domain Visco-Elastic Wave Equations in 3D Heterogeneous Media: A Controllability Approach

Fully Scalable Solver for Frequency-Domain Visco-Elastic Wave Equations in 3D Heterogeneous Media: A Controllability Approach

Global existence and new decay results of a viscoelastic wave equation with variable exponent and logarithmic nonlinearities

<abstract> In this paper, we consider the following viscoelastic problem with variable exponent and logarithmic nonlinearities: <p class="disp_formula">$ u_{tt}-\Delta u+u+ \int_0^tb(t-s)\Delta u(s)ds+|u_t|^{{\gamma}(\cdot)-2}u_t = u\ln{\vert u\vert^{\alpha}}, $ where $ {\gamma}(.) $ is a function satisfying some conditions. We first prove a global existence result using the well-depth method and then establish explicit and general decay results under a wide class of relaxation functions and some specific conditions on the variable exponent function. Our results extend and generalize many earlier results in the literature. </abstract>

Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation

<p style='text-indent:20px;'>We are concerned with the transmission problem of nonlinear viscoelastic waves in a heterogeneous medium, establishing the well-posedness of solutions and the exponential stability of the related energy functional. We introduce an auxiliary problem to prove the exponential stability and the proof combines an observability inequality and microlocal analysis tools.

On the decay of solutions of a viscoelastic wave equation with variable sources

In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (smart fluids), which show changing (often dramatically) in the viscosity when an electrical field is applied. The Lebesgue and Sobolev spaces with variable exponents are efficient tools to analyze such problems. In this work, we prove a global existence result using the well‐depth method and establish explicit and general decay results under a very general assumption on the relaxation function. Our results extend and generalize many results in the literature.

General decay result of solutions for viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term

In this paper, we consider a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term. Under suitable assumptions on relaxation functions, we establish general decay result for energy.

New general decay result for a system of viscoelastic wave equations with past history

<p style='text-indent:20px;'>This work is concerned with a coupled system of viscoelastic wave equations in the presence of infinite-memory terms. We show that the stability of the system holds for a much larger class of kernels. More precisely, we consider the kernels <inline-formula><tex-math id="M1">\begin{document}$ g_i : [0, +\infty) \rightarrow (0, +\infty) $\end{document}</tex-math></inline-formula> satisfying <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ g_i'(t)\leq-\xi_i(t)H_i(g_i(t)),\qquad\forall\,t\geq0 \quad\mathrm{and\ for\ }i = 1,2, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H_i $\end{document}</tex-math></inline-formula> are functions satisfying some specific properties. Under this very general assumption on the behavior of <inline-formula><tex-math id="M4">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity, we establish a relation between the decay rate of the solutions and the growth of <inline-formula><tex-math id="M5">\begin{document}$ g_i $\end{document}</tex-math></inline-formula> at infinity. This work generalizes and improves earlier results in the literature. Moreover, we drop the boundedness assumptions on the history data, usually made in the literature.

Dynamics of a coupled system for nonlinear damped wave equations with variable exponents

AbstractWe consider a coupled system of viscoelastic wave equation with weak, strong damping and power nonlinearity. For a single viscoelastic wave equation, we have already obtained a global solution, its decay rate and finite‐time blow‐up [1]. In this paper, we extend these results to a coupled system. First, we obtain a global solution and derive its decay rate by a decreasing energy. Finally, we apply the concavity method in order to show that the solution blows up in finite time under nonclassic constraint on kernel functions.

Application of the Crank–Nicolson time integrator to viscoelastic wave equations with boundary feedback damping

Abstract We study the numerical solutions for a boundary feedback mechanism on torsional vibrations of a homogeneous viscoelastic rod. The problem is discretized by the Crank–Nicolson scheme based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature formula, constructed again from the trapezoidal rule, is used to approximate the integral term. Error estimates of the numerical scheme are derived in the $ l^{\infty }_{t}(0,~\infty ;~L^{2}(0,~1)) $ norm. Results from some numerical experiment are presented.

A sequential inversion with outer iterations for the velocity and the intrinsic attenuation using an efficient wavefield inversion

Full-waveform inversion (FWI) has become a popular method to retrieve high-resolution subsurface model parameters. It is a highly nonlinear optimization problem based on minimizing the misfit between the observed and predicted data. For intrinsically attenuating media, wave propagation experiences significant loss of energy. Thus, for better data fitting, it is sometimes crucial to consider attenuation in FWI. Viscoacoustic FWI aims at achieving a joint inversion of the velocity and attenuation models. However, multiparameter FWI imposes additional challenges including expanding the null space and facing parameter trade-off issues. Theoretically, an ideal way to mitigate the trade-off issue in multiparameter FWI is to apply the inverse Hessian operator to the parameter gradients. However, it is often not practical to calculate the full Hessian and its matrix inverse because this will be extremely expensive. To improve the computational efficiency and mitigate the trade-off issue, we have used an efficient wavefield inversion (EWI) method to invert for the velocity and the intrinsic attenuation. This approach is implemented in the frequency domain, and the velocity, in this case, is complex-valued in the viscoacoustic EWI. We evaluate a sequential update strategy for the velocity and the intrinsic attenuation, and we repeat the separate optimizations, which we refer to as outer iterations, until the convergence is achieved. Because viscoacoustic EWI is able to recover an accurate velocity model, the velocity update leakage to the [Formula: see text] model is largely reduced. We determine the effectiveness of this approach using synthetic data generated for the viscoacoustic Marmousi and Overthrust models. To further demonstrate the validity of our approach, we generate data in the time domain using a viscoelastic wave equation solver and obtain reasonable inversion results in the frequency domain using the viscoacoustic approximation.

On the General Stability of a Viscoelastic Wave Equation with an Integral Condition

We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.

Exponential decay of the viscoelastic wave equation of Kirchhoff type with a nonlocal dissipation

The viscoelastic wave equation of Kirchhoff type with nonlinear and nonlocal damping is considered in a bounded domain of R^N. The existence of global solutions and decay rates of the energy are proved.

Energy decay for variable coefficient viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary

In this article, we study a variable coefficients viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary. By constructing suitable Lyapunov functionals and using the energy compensation method, we prove that under suitable conditions on the initial data and the relaxation function, the energy of the system has an explicit and general decay rate.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/95/abstr.html

Viscoelastic Wave Simulation with High Temporal Accuracy Using Frequency-Dependent Complex Velocity

In recent decades, the study of seismic attenuation has received more and more concerns because it can stimulate the development of wave propagation simulation and improve the accuracy of structure imaging and reservoir prediction. In this paper, we review the attenuation theory and the development of high temporal accuracy wave simulation. The conventional mathematical models to describe the characteristics of viscoelastic are based on constant-Q model or standard linear solids theory. However, these approaches possess some noticeable shortcomings. Therefore, we introduce a frequency-dependent complex velocity to derive the novel viscoelastic wave equations with decoupled amplitude dissipation and phase dispersion. To obtain high temporal accuracy viscoelastic wave simulation, we adopt the normalized pseudo-Laplacian to compensate for the temporal dispersion errors caused by the second-order finite-difference discretization in the time domain. During the implementation, we incorporate the normalized pseudo-Laplacian into the optimized staggered-grid finite-difference coefficients. Therefore, it can greatly reduce the times of low-rank decomposition and Fourier transform and largely improve the computational efficiency. Based on this strategy, we can implement the high temporal accuracy viscoelastic wavefield extrapolation by comprehensively exploiting the staggered-grid finite-difference scheme, pseudo-spectral method and low-rank decomposition algorithm. Meanwhile, a linear velocity model is employed to evaluate the accuracy of low-rank approximation. Furthermore, we use several numerical examples to carry out the comparison between our scheme and other conventional methods. The numerical results reveal that our proposed scheme can effectively compensate for temporal dispersion errors and help generate high temporal accuracy viscoelastic wave solutions.

A Splitting Mixed Covolume Method for Viscoelastic Wave Equations on Triangular Grids

A new splitting mixed covolume (SMCV) method is proposed for the viscoelastic wave equations by combining the splitting positive definite mixed finite element (SPDMFE) method with the mixed covolume (MCV) method. Based on the idea of the SPDMFE method, the difference between this method and the MCV method is that the proposed method does not need to solve the coupled system, thus reducing the scale of linear equations. By introducing a transfer operator, the semi-discrete and fully discrete SMCV schemes are proposed. The existence and uniqueness analysis of the semi-discrete scheme are given, and optimal a priori error estimates in different norm for these two schemes are derived. Finally, the feasibility and effectiveness of the proposed scheme are verified by some numerical results.

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.

Global existence, nonexistence, and decay of solutions for a viscoelastic wave equation with nonlinear boundary damping and source terms

This paper deals with a high order viscoelastic wave equation with nonlinear boundary damping–source interactions. By the combination of Galerkin approximation and potential well methods, we obtain the global existence of weak solutions. Under some appropriate assumptions on the boundary damping–source terms and the relaxation function g, we establish general decay and blow-up results associated with solution energy. Estimates of the lifespan of solutions are also given.

Exponential growth of solution with \(L_{p}\) -norm for class of non-linear viscoelastic wave equation with distributed delay term for large initial data

Exponential growth of solution with \(L_{p}\) -norm for class of non-linear viscoelastic wave equation with distributed delay term for large initial data

Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form utt(x,t)−Δu(x,t)+∫0tg(t−s)Δu(x,s)ds−Δut(x,t)=|u(x,t)|p−2u(x,t)ln|u(x,t)|\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}(x,t) - \\Delta u (x,t) + \\int ^{t}_{0} g(t-s) \\Delta u(x,s)\\,ds - \\Delta u_{t} (x,t) = \\bigl\\vert u(x,t) \\bigr\\vert ^{p-2} u(x,t) \\ln \\bigl\\vert u(x,t) \\bigr\\vert $$\\end{document} in a bounded domain varOmega subset {mathbb{R}}^{n} , where g is a nonincreasing positive function. Firstly, we prove the existence and uniqueness of local weak solutions by using Faedo–Galerkin’s method and contraction mapping principle. Then we establish a finite time blow-up result for the solution with positive initial energy as well as nonpositive initial energy.