Well-Posedness and Asymptotic Analysis of Wave Equation with Nonlocal Boundary Damping

<p style='text-indent:20px;'>The viscoelastic wave equation with nonlinear nonlocal weak damping is considered. The local existence of solutions is established. Under arbitrary positive initial energy, a finite-time blow-up result is proved by a new modified concavity method.</p>

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical ( ) and critical ( ) initial energy levels. We also prove the finite time blow‐up of the solution for the initial data at arbitrary high energy level, including the estimates of lower and upper bounds of the blow‐up time.

General Energy Decay of Solutions for a Wave Equation with Nonlocal Damping and Nonlinear Boundary Damping

In this paper, we study the long-time dynamics for the autonomous wave equation with nonlocal weak damping and super-cubic nonlinearity in a bounded smooth domain of R3. Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah–Struwe solutions. Then we establish the existence of the global attractor for the Shatah–Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.

This work is devoted to studying a wave equation with degenerate nonlocal nonlinear damping and source terms. By potential well theory, we show the asymptotic stability of energy in the presence of a degenerate damping of polynomial type when the initial energy is small. Also, we firstly derive some sufficient conditions on initial data which lead to finite time blow-up.

<p style='text-indent:20px;'>In this paper, we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.</p>

A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.

In this paper, we consider a nonlinear viscoelastic equation with nonlocal nonlinear boundary damping and two nonlinear source terms (boundary and interior). We obtain the global existence of solution via the potential well method. By introducing suitable Lyapunov functionals, using the multiplier method, and constructing convex differential inequality, we establish an explicit and general decay rate result. This improves previous decay results concerning the problem. We also get a finite time blow-up result of solution with positive initial energy under suitable conditions on the initial data and positive memory function. This is the first result for blow-up of the problem with nonlocal nonlinear boundary damping.

This article presents a new scheme for studying the dynamics of a quintic wave equation with nonlocal weak damping in a 3D smooth bounded domain. As an application, the existence and structure of weak, strong, and exponential attractors for the solution semigroup of this equation are obtained. The investigation sheds light on the well-posedness and long-term behavior of nonlinear dissipative evolution equations with nonlinear damping and critical nonlinearity.

Existence of a generalized polynomial attractor for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity

Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping

<p style='text-indent:20px;'>The paper is devoted to establishing the long-time behavior of solutions for the wave equation with nonlocal strong damping: <inline-formula><tex-math id="M1">\begin{document}$ u_{tt}-\Delta u-\|\nabla u_{t}\|^{p}\Delta u_{t}+f(u) = h(x). $\end{document}</tex-math></inline-formula> It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is up to the subcritical and critical cases in natural energy space.</p>

Exponential Attractors for the Sup-Cubic Wave Equation with Nonlocal Damping

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.

A parameter study was done to determine what errors may exist in normal wave equation analysis parameters which would not affect results for short piles but which might give significantly erroneous results for very long piles. The only wave equation analysis parameter which affects energy losses for the portion of piles extending through water is called the "coefficient of restitution" (eras) of the pile. The normal value of Ares used for pile segments in the wave equation analysis is 1.0. This value of Ares means that the pile is assumed to be perfectly elastic and to have no energy losses. It was found by this parameter study that a slightly lower value of eras used for the pile segments had very little effect on calculated static pile capacity for piles up to a length of about 150 ft. For very long piles, however, the calculated ultimate capacity was significantly reduced. INTRODUCTION The wave equation analysis of pile driving was developed by E.A.L. Smith and first presented to the civil engineering profession in 1960, Smith3 At the time it was developed, certain standard parameters were used in the wave equation analysis which were found to give correct results for the piles commonly driven in the years from 1940 to 1960 or so. The majority of these piles were from 30 ft. to 60 ft. long and carried design loads from 15 tons to 60 tons. In the years since 1960, the trend has been toward longer and higher capacity driven piles. The wave equation analysis has been used essentially without modification to analyze the driving of these piles. The longest and heaviest of these piles are, of course, the piles used in offshore work. Even today, most of the standard parameters recommended by Ed Smith in 1960 are still used in most of the wave equation analyses which are run. Examples of these standard parameters which are routinely used in most wave equation analyses are as follows:(Available in full paper) Ed Smith proposed that the above standard values be used until more accurate values became available. He justified the use of these values even though they were admittedly not accurate because ..."the numerical wave equation solution is not 'sensitive', that is, a small change in the value assigned to any constant will produce a smaller change in the calculated results."3 For piles in the general length and capacity range of those days, this statement was correct. At the present time, however, especially for the very large diameter and very long piles used in the offshore construction industry, all of these standard parameters should be re-examined to see if the wave equation solution is possibly "sensitive" to certain parameters for very long piles.