Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.     

Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves

In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects the energy of solutions obtained by the PML method to tend to zero as t → ∞, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that, due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements on regular meshes and to multi-dimensional problems. This work started while the first author was visiting the Department of Mathematics of the Universidad Autónoma de Madrid, in the frame of the European program “New materials, adaptive systems and their nonlinearities: modeling, control and numerical simulation” HPRN-CT-2002-00284. The work was finished while both authors visited the Isaac Newton Institute of Cambridge within the Program “Highly Oscillatory Problems”.     

Asymptotic behavior of solutions for initial–boundary value problems for strongly damped nonlinear wave equations

We study initial–boundary value problems for strongly damped nonlinear wave equations. By using improved integral estimates, it is proven that the solutions of the problems decay to zero exponentially as time t$t$ approaches infinity, under a very simple and general assumption regarding the nonlinear term.     

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.     

Large‐time behavior of solutions to the Rosenau equation with damped term

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping

In this paper we consider the following Timoshenko system: with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ₁, ρ₂, k₁, k₂ and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Well‐posedness for the 2‐D damped wave equations with exponential source terms

In this paper we study well‐posedness of the damped nonlinear wave equation in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ?²; ω?0, ωλ₁+µ>0 with λ₁ being the first eigenvalue of ?Δ under zero boundary condition. Under the assumptions that g(·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow‐up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.     

Discrete versus continuous Newton's method: A case study

We consider the damped Newton's method N _h (z) = z – hp(z)/p(z), 0<h<1 for polynomialsp(z) with complex coefficients. For the usual Newton's method (h=1) and polynomialsp(z), it is known that the method may fail to converge to a root ofp and rather leads to an attractive periodic cycle.N _h(z) may be interpreted as an Euler step for the differential equation =–p(z)/p(z) with step sizeh. In contrast to the possible failure of Newton's method, we have that for almost all initial conditions to the differential equation that the solutions converge to a root ofp. We show that this property generally carries over to Newton's methodN _h(z) only for certain nondegenerate polynomials and for sufficiently small step sizesh>0. Further we discuss the damped Newton's method applied to the family of polynomials of degree 3.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

Limiting dynamics for stochastic wave equations

In this paper, relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. By introducing almost surely D-α-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation endowed with Dirichlet boundary condition for any 0<ν?1. The upper semicontinuity of this global random attractor and the global attractor of the heat equation zt−Δz+f(z)=0 with Dirichlet boundary condition as ν goes to zero is investigated. Furthermore we show the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Uniformly Controllable Schemes for the Wave Equation on the Unit Square

The paper deals with the numerical approximation of the HUM control of the 2D wave equation. Most of the discrete models obtained with classical finite difference or finite element methods do not produce convergent sequences of discrete controls, as the mesh size h and the time step Δt go to zero. We introduce a family of fully-discrete schemes, nondispersive, stable under the condition \(\Delta t\leq h\slash\sqrt{2}\) and uniformly controllable with respect to h and Δt. These implicit schemes differ from the usual explicit one (obtained with leapfrog time approximation and five point spatial approximations) by the addition of terms proportional to h ² and Δt ². Numerical experiments for nonsmooth initial conditions on the unit square using a conjugate gradient algorithm indicate the excellent performance of the schemes.     

Oscillation of third order differential equation with damping term

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term

$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$

We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Stabilization and control for the subcritical semilinear wave equation

In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cut-off away from the boundary.     

Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions

A variable coefficient viscoelastic wave equation with acoustic boundary conditions and nonlinear source term is considered. Under suitable conditions on the initial data and the relaxation function g, we show the polynomial decay of the energy solution and the blow up of solutions by energy methods. The estimates for the lifespan of solutions are also given.     

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

We consider the Cauchy problem in R ⁿ for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L^1,1( R ⁿ) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L²‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.     

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially even if m > 2.