Damped Wave Equation Research Articles

Arbitrary initial energy blow up for fourth-order viscous damped wave equation with exponential-type growth nonlinearity

This letter is concerned with the following damped nonlinear wave equation with exponential nonlinearity utt+Δ2u+u+Δ2ut+ut=f(u)inΩ×(0,∞)where the nonlinearity f is a regular function satisfying f(0)=0 with an exponential growth to fix later. The above wave problem describes a class of essential nonlinear evolution equations appearing in the elastic–plastic-microstructure models. In this letter blow up result is established for arbitrary initial energy with the viscous damping terms.

Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor {A_{varepsilon }(t)}_{tin mathbb{R}} of Eq. (1.1) with varepsilon in [0,1] satisfies lim_{varepsilon to varepsilon _{0}}sup_{tin [a,b]} operatorname{dist}_{H_{0}^{1}times L^{2}}(A_{varepsilon }(t),A_{ varepsilon _{0}}(t))=0 for any [a,b]subset mathbb{R} and varepsilon _{0}in [0,1].

Exponential stability of a pendulum in dynamic boundary feedback with a viscous damped wave equation

In this paper, we continue the earlier work [Lu, L., & Wang, D. L. (2017). Dynamic boundary feedback of a pendulum coupled with a viscous damped wave equation. In Proceedings of the 36th Chinese Control Conference(CCC) (pp. 1676–1680)] on study the stability of a pendulum coupled with a viscous damped wave equation model. This time we get the exponential stability result which is much better than the previous strong stability. By a detailed spectral analysis and operator separation, we establish the Riesz basis property as well as the spectrum determined growth condition for the system. Finally, the exponential stability of the system is achieved.

Convergence of Non-autonomous Attractors for Subquintic Weakly Damped Wave Equation

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah–Struwe solutions, which satisfy the Strichartz estimates and coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, uniform, and cocycle attractors and the relations between them. We also prove that these non-autonomous attractors converge upper-semicontinuously to the global attractor for the limit autonomous problem if the time-dependent nonlinearity tends to a time independent function in an appropriate way.

Unconditional Energy Dissipation and Error Estimates of the SAV Fourier Spectral Method for Nonlinear Fractional Generalized Wave Equation

In this paper, we consider a second-order scalar auxiliary variable Fourier spectral method to solve the nonlinear fractional generalized wave equation. Unconditional energy conservation or dissipation properties of the fully discrete scheme are first established. Next, we utilize the temporal-spatial error splitting argument to obtain unconditional optimal error estimate of the fully discrete scheme, which overcomes time-step restrictions caused by strongly nonlinear system, or the restrictions that the nonlinear term needs to satisfy the assumption of global Lipschitz condition in all previous works for fractional undamped or damped wave equations. Finally, some numerical experiments are presented to confirm our theoretical analysis.

Network-based deployment of nonlinear multi agents over open curves: A PDE approach

Deployment of the first-order and second-order nonlinear multi agent systems over desired open (and, as a particular case, closed) smooth curves in 2D or 3D space is considered. The considered nonlinearities are globally Lipschitz. We assume that the agents have access to the local information of the desired curve and to their positions with respect to their closest neighbors (as well as to their velocities for the second-order systems), whereas in addition a leader agent is able to measure its absolute position. We assume that a small number of leaders (distributed in the spatial domain) transmit their measurements to other agents through a communication network. We take into account the following network imperfections: variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear heat equation (for the first-order systems) or damped wave equation (for the second-order systems) with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. In order to cope with the open curve we consider Neumann boundary conditions that ensure mobility of the boundary agents. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical examples illustrate the efficiency of the method.

Fujita modified exponent for scale invariant damped semilinear wave equations

The aim of this paper is to prove a blow-up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition, we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term and not only on the damping coefficient but also on the size of the decay rate of the initial data.

A rigorous derivation and energetics of a wave equation with fractional damping

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water–air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.

Existence of random attractors for strongly damped wave equations with multiplicative noise unbounded domain

In this paper, we establish the existence of a random attractor for a random dynamical system generated by the non-autonomous wave equation with strong damping and multiplicative noise when the nonlinear term satisfies a critical growth condition.

New Estimates of Solution to Coupled System of Damped Wave Equations with Logarithmic External Forces

In the paper, we consider new stability results of solution to class of coupled damped wave equations with logarithmic sources in ℝ n . We prove a new scenario of stability estimates by introducing a suitable Lyapunov functional combined with some estimates.

Initial boundary value problem for a strongly damped wave equation with a general nonlinearity

<p style='text-indent:20px;'>In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.</p>

On the inverse problem for nonlinear strongly damped wave equations with discrete random noise

Abstract In this paper, we consider the existence of a solution u(x, t) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q (q > 0) norms. Moreover, a numerical example is given to illustrate our results.

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.

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

Blow-up results for semilinear damped wave equations in Einstein–de Sitter spacetime

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein–de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural generalization of Strauss exponent. In the critical case, we consider a non-autonomous and parameter dependent Cauchy problem for a linear ODE of second order, whose explicit solutions are determined by means of special functions’ theory.

Damped quantum wave equation from non-standard Lagrangians and damping terms removal

Hyperbolic equations are used in several physical phenomena to describe dynamical processes where information propagates with a finite speed. Recently, the wave equations with damping terms turned out to be fundamental hyperbolic equations in certain branches of physics mainly scattering processes and fractal medium. The aim of the present study is double shooting, first to prove that damped quantum wave equations may be obtained using the notion of non-standard Lagrangians and second to show that linear and nonlinear damping terms may be obtained if the concept of ‘two-occurrences of integrals’ is used, hence reducing the damped quantum wave equation to the conventional quantum wave equation known as the Klein-Gordon equation. This study supports the idea of non-standard Lagrangians and its usefulness in the theory of partial differential equations.

Rapid Exponential Stabilization of Nonlinear Wave Equation Derived from Brain Activity via Event-Triggered Impulsive Control

This paper investigates the problem of rapid exponential stabilization for damped wave equations. Based on a new event-triggered impulsive control (ETIC) method, impulsive control was designed to solve the rapid exponential stabilization of a class of damped wave equations derived from brain activity. The effectiveness of our control was verified through a numerical example.

Infinite energy solutions for weakly damped quintic wave equations in ℝ³

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

Some inverse problems for wave equations with fractional derivative attenuation

The damped wave equation with the attenuation proportional to velocity is ubiquitous in science and engineering and a common situation is when the attenuation depends on frequency. The usual way to incorporate this effect is to introduce fractional order derivatives either as a replacement for u t or as modifier through a spatial component with space fractional derivatives. Models for these are very well developed and the effort in this paper is towards the analysis of the inverse problem of recovering critical coefficients or initial states although we also develop constructive methods for these and analyse their degree of ill-conditioning.

Numerical simulation based heuristic investigation of inertia-induced phenomena and theoretical solution based verification by the damped wave equation for the dynamic deformation of saturated soil based on the u–w–p governing equation

The theoretical and numerical solutions for the one-dimensional dynamic deformation of saturated ground were obtained using the u–w–p governing equation. The numerical analysis was performed based on the finite element method, and the occurrence of superficially peculiar phenomena, i.e., S-shaped settlement–time relation, non-harmonic oscillation in the settlement/pressure–time relation in highly permeable soil, and inconsistency in the vertical load and immediate pressure was identified heuristically. The theoretical solution of the damped wave equation was derived from the original governing equation expressed as the superposition of several modes with different eigenvalues, and this solution could successfully explain the phenomena observed heuristically. It was noted that the phenomena were induced by the presence of inertia, because the static solution and u–p analysis did not exhibit such phenomena. The verification of the analysis was conducted owing to the correspondence of the theoretical solutions of the damped wave equation and the u–w–p analysis results.