Nonlinear Dissipative Term Research Articles

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term

The Cauchy-Goursat problem for wave equations with nonlinear dissipative term is studied. The existence, uniqueness, and blow-up of global solutions of this problem are considered. The local solvability of this problem is also discussed.

Parametric Stability and Bifurcations of Axially Moving Viscoelastic Beams with Time-Dependent Axial Speed#

The subject of this article is the investigation of the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed. The force and moment balances as well as constitutive relations are employed to derive the equation of motion. Due to the presence of the time-dependent axial speed and steady dissipation terms, time-dependent coefficients and nonlinear dissipation terms are generated, respectively. The equation of motion is reduced into a set of coupled nonlinear ordinary differential equations with time-dependent coefficients. The subcritical resonant response of the system is obtained using the pseudo-arclength continuation technique, while the bifurcation diagrams of Poincaré maps are obtained via direct time integration of the discretized equations of motion.

Energy decay for a nonlinear generalized Klein-Gordon equation in exterior domains with a nonlinear localized dissipative term

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation in exterior domains with a nonlinear localized dissipation. Our equation includes an absorbing term like |u|αu, α ≥ 0, and can be regarded as a generalized Klein-Gordon equation at least if α is closed to 0. This observation plays an essential role in our argument.

Dissipative perturbations for the [formula omitted] Rosenau–Hyman equation

Compactons are compactly supported solitary waves for nondissipative evolution equations with nonlinear dispersion. In applications, these model equations are accompanied by dissipative terms which can be treated as small perturbations. We apply the method of adiabatic perturbations to compactons governed by the K(n,n) Rosenau–Hyman equation in the presence of dissipative terms preserving the “mass” of the compactons. The evolution equations for both the velocity and the amplitude of the compactons are determined for some linear and nonlinear dissipative terms: second-, fourth-, and sixth-order in the former case, and second- and fourth-order in the latter one. The numerical validation of the method is presented for a fourth-order, linear, dissipative perturbation which corresponds to a singular perturbation term.

The first Darboux problem for wave equations with nonlinear dissipative term

The first Darboux problem for wave equations with nonlinear dissipative term is considered. The uniqueness, local and global existence and blow-up of solutions of the problem mentioned are investigated. The paper’s originality is the coalescence of two standard methods: a priori estimate of solutions in the class of continuous functions is given by energetic methods; basing on this result a priori estimate in the class of continuously differentiable functions using classical method of characteristics is obtained.

Classical open systems with nonlinear nonlocal dissipation and state-dependent diffusion: Dynamical responses and the Jarzynski equality

We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system (x) to applied sinusoidal and step forces significantly depend on model parameters of magnitudes of additive and multiplicative noises and the relaxation time of colored noise, although stationary marginal probability distribution functions are independent of them, (b) a combined effect of nonlinear dissipation and multiplicative colored noise induces enhanced fluctuations ([x-(x)](2)) for an applied sinusoidal force, and (c) the JE holds for an applied ramp force independently of the model parameters with a work distribution function which is (symmetric) Gaussian and (asymmetric) non-Gaussian for additive and multiplicative noises, respectively. It has been shown that the non-Markovian Langevin equation in the local and overdamped limits is quite different from the widely adopted, phenomenological Markovian Langevin equation subjected to additive and multiplicative noises.

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H 2 solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.

Energy decay to the Cauchy problem for a generalised nonlinear Klein-Gordon equation with a nonlinear dissipative term

We derive a precise decay rate of energy of solutions to the Cauchy problem for the wave equation with a nonlinear dissipative term and an absorbing term. Due to the absorbing term the equation is regarded as a generalised Klein-Gordon equation with a nonlinear dissipative term. This observation plays an important role in the proof.

The boundary value problem for wave equations with nonlinear dissipative and source terms

The boundary value problem for wave equations with nonlinear dissipative and source terms in an angular domain is considered. The questions of existence and non-existence of global and blow-up solutions of considering problem are investigated for some values of parameters in the nonlinear terms and the right-hand side of given equations. The uniqueness, local and global existence and blow-up of solutions of the problem mentioned are investigated.

Generation mechanisms for capillary-gravity wind wave spectrum

We investigate the role of different physical mechanisms in the generation of the capillary-gravity wind wave spectrum. This spectrum is calculated by integrating a nonstationary kinetic equation until the solution becomes stready. The mechanisms of spectrum generation under consideration include three-wave interactions, viscous dissipation, energy influx from wind, nonlinear dissipation, and the generation of a parasitic capillary ripple. The three-wave interactions are taken into account as an integral of collisions without additional simplifications. It is shown that the three-wave interactions lead to solution instability if the kinetic equation takes into account only linear sources. To stabilize the solution, the kinetic equation should incorporate a nonlinear dissipation term, which in the range of short gravity waves corresponds to energy losses during wave breaking and microscale wave breaking. In the range of capillary waves, the account of nonlinear dissipation is also needed to ensure a realistic level of the spectrum for large wind velocities. For the steady-state spectrum, the role of three-wave interactions remains essential merely in the range of the minimum of phase velocity, where a trough on the curvature spectrum is formed. At the remaining intervals of the spectrum, the main contribution into the spectral energy balance is provided by the mechanisms of wave injection, nonlinear dissipation, and the generation of parasitic capillaries.

Remarks on the energy decay for nonlinear wave equations with nonlinear localized dissipative terms

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation with a nonlinear localized dissipation. To overcome a difficulty related to derivative-loss mechanism we employ a ‘loan’ method.

Global attractor for a class of nonlinear lattices

We consider a class of nonlinear lattices with nonlinear damping (0.1) u ¨ n ( t ) + ( − 1 ) p Δ p u n ( t ) + α u n ( t ) + h ( u n ( t ) ) + g ( n , u ˙ n ( t ) ) = f n , where n ∈ Z , t ∈ R + , α is a real positive constant, p is any positive integer and Δ is the discrete one-dimensional Laplace operator. Under suitable conditions on h and g we prove the existence of a global attractor for the continuous semigroup associated with (0.1). Our proofs are based on a difference inequality due to M. Nakao [M. Nakao, Global attractors for nonlinear wave equations with nonlinear dissipative terms, J. Differential Equations 227 (2006) 204–229].

A hybrid legendre tau method for the solution of a class of nonlinear wave equations with nonlinear dissipative terms

This article presents a technique based on the hybrid Legendre tau-finite difference method to solve the fourth order wave equation which arises in the elasto-plastic-microstructure models for longitudinal motion of an elasto-plastic bar. Illustrative examples and numerical results obtained using new technique demonstrate that the proposed approach is efficient in treating longitudinal equation of ealsto-plastic bar. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1055–1071, 2011

Global Existence and Asymptotic Behavior of Solutions for Some Nonlinear Hyperbolic Equation

The initial boundary value problem for a class of hyperbolic equation with nonlinear dissipative term in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in and show the asymptotic behavior of the global solutions through the use of an important lemma of Komornik.

General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms

A viscoelastic wave equation in canonical form with weakly nonlinear time-dependent dissipation and source terms is investigated in this paper. For a wider class of relaxation functions and without imposing any restrictive growth assumption on the damping term at the origin, we establish an explicit and general energy decay rate result.

On the strongly damped wave equation with nonlinear damping and source terms

We consider a wave equation in a bounded domain with nonlinear dissipation and nonlinear source term. Characterizations with respect to qualitative properties of the solution: globality, boundedness, blow-up, convergence up to a subsequence towards the equilibria and exponential stability are given in this article.

Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipation

In this paper we prove the existence of global decaying H2 solutions to the Cauchy problem for a wave equation with a nonlinear dissipative term by constructing a stable set in H1(ℝn). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The generalized BBM-Burger equations with non-linear dissipative term: existence and convergence results

We study the global existence of solutions for certain equations of the form u t + f(u) x = γB(u x ) x + δu xxt − αu xxxx , as γ > 0, δ > 0 and α > 0 aproach zero, and f and B are sufficiently smooth functions satisfying certain appropriate assumptions. We consider solutions of hyperbolic conservation laws regularized of this equations. Following a pioneering work by Schonbek and a work by LeFloch and Natalini, we establish the convergence of the regularized solutions towards discontinuous solutions of the hyperbolic conservation law.

Decay Estimates of Solutions to the Cauchy Problem for a Wave Equation with a Bounded Nonlinear Dissipation

In this paper we prove the global existence and study decay properties of the solutions to the Cauchy problem for a wave equation with a bounded nonlinear dissipative term.

Proposal of particle swarm optimization methods with nonlinear dissipative term

AbstractOptimization methods based on metaheuristics are proposed as a class of global optimization methods, by which the global minimum can be obtained without being trapped in local minima. Particle swarm optimization (PSO), which is one of these methods, is known for its high searching ability and easy implementation. However, it might be difficult to find the global optimum for optimization problems with a number of decision variables and multiple local optima. In this paper, we propose three types of new PSO methods to overcome this difficulty. One is a model with nonlinear dissipative term introduced by Fujita and colleagues [4] to prohibit the search point's velocity from being zero. The others are models with the nonlinear dissipative term with the pbest or the gbest information to disturb the search around them. © 2008 Wiley Periodicals, Inc. Electron Comm Jpn, 91(2): 23–30, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10028