Nonlinear Dissipative Wave Equations Research Articles

Application of variational iteration method to non-homogeneous non-linear dissipative wave equations

In this paper, the solution of the non-linear non-homogeneous dissipative wave equation is obtained by means of the variational iteration method. This equation describes the propagation of a wave, and it arises in a wide variety of physical problems. The results reveal that the variational iteration method is very effective, convenient and quite accurate to systems of nonlinear partial differential equations.

Local energy decay for the nonlinear dissipative wave equation in an exterior domain

In odd dimension space, under a microlocal geometric condition, we give the rate of decay of the local energy for solutions of the wave equation on exterior domain, with localized nonlinear damping.

Blow-up of solutions of a class of strongly non-linear dissipative wave equations of Sobolev type with sources

We consider the abstract Cauchy problem for a first-order ordinary differential equation with non-linear operator coefficients. The results are applied to some strongly non-linear dissipative wave equations of Sobolev type. We obtain sufficient conditions for the problem to be globally soluble, as well as sufficient conditions for the solutions to blow up in finite time. These conditions are close to being necessary. Under certain supplementary assumptions on the non-linear operators, we prove that the problem is soluble in any finite cylinder. Under certain conditions on the norm of the initial functions, we prove that the solution of the problem blows up in finite time. We give examples of equations of Sobolev type satisfying these conditions.

Blow-up for nonlinear dissipative wave equations in [formula omitted

We consider the Cauchy problem { u t t − Δ u + | u t | m − 1 u t = | u | p − 1 u , ( t , x ) ∈ ( 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = v 0 ( x ) , for 1 ⩽ m < p , p < n / ( n − 2 ) for n ⩾ 3 . We prove that for any given numbers α > 0 , λ ⩾ 0 there exist infinitely many data u 0 , v 0 in the energy space such that the initial energy E ( 0 ) = λ , the gradient norm ‖ ∇ u 0 ‖ 2 = α , and the solution of the above Cauchy problem blows up in finite time.

Kolmogorov ε-Entropy in the Problems on Global Attractors for Evolution Equations of Mathematical Physics

We study the Kolmogorov e-entropy and the fractal dimension of global attractors for autonomous and nonautonomous equations of mathematical physics. We prove upper estimates for the e-entropy and fractal dimension of the global attractors of nonlinear dissipative wave equations.

Asymptotics for the nonlinear dissipative wave equation

We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like α 1 (1+|x|) -δ |u t | β u t (α 1 , β, δ > 0) and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

An application of the decomposition method for second order wave equations

In this paper we study the solution of a linear and nonlinear damped wave and dissipative wave equations by Adomian decomposition method. We illustrate that the analytic solutions and a reliable numerical approximation of the damped wave and dissipative wave equations are calculated in the form of a series with easily computable components. The nonhomogeneous problem is quickly solved by observing the self-canceling"noise"terms whose sum vanishes in the limit. In comparison to traditional techniques, the series based technique of Adomian decomposition method is shown to evaluate solutions accurately and cheaply.

Nonlinear dissipative wave equations

The decomposition method is applied to nonlinear dissipative wave equations.

Asymptotic property for the solution to the generalized Korteweg-de vries equation

We study temporal asymptotic property for the solution of the initial value problem for the generalized nonlinear Korteweg-de Vries equation, which is a nonlinear dissipative dispersive wave equation. The decay results of the solution follow from thepriori L2 integral estimates and the Fourier transform.

Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations

Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations

Application of decomposition to hyperbolic, parabolic, and elliptic partial differential equations

The decomposition method is applied to examples of hyperbolic, parabolic, and elliptic partial differential equations without use of linearizatlon techniques. We consider first a nonlinear dissipative wave equation; second, a nonlinear equation modeling convectlon‐diffusion processes; and finally, an elliptic partial differential equation.

Addendum to: Existence and uniqueness of solutions to nonlinear dissipative wave equations

Addendum to: Existence and uniqueness of solutions to nonlinear dissipative wave equations

Existence and uniqueness of solutions to nonlinear dissipative wave equations

On etudie l'existence et l'unicite des solutions d'equations d'ondes dissipatives non lineaires avec donnees de Cauchy dans l'espace de mesure. Les donnees de Cauchy ont une energie infinie, ce qui ne permet pas d'utiliser la methode d'energie classique

On the decay of solutions of some nonlinear dissipative wave equations in higher dimensions

On etudie la propriete de decroissance des solutions des equations d'onde dissipatives non lineaires de la forme: ∂ 2 u/∂t 2 -Δu+g(∂u/∂t)=f(x,t), (x,t)∈Ω×R + avec les conditions u(x,o)=u 0 (x), u t (x,o)=u 1 (x), x∈Ω et u(x,t)=0 sur ∂Ω×R + , ou Ω est un domaine borne de R n avec une frontiere lisse ∂Ω et g(v) est une fonction du genre k/v/ α v,k>o,α≥o

Amplitude Dispersion and Stability of Nonlinear Weakly Dissipative Waves

A method to solve nonlinear dissipative wave equations using ideas of Luke, Krylov, and Bogolyubov is presented. The method is compared to Whitham's theory. Dispersion relations for nonlinear dissipative waves, including amplitude dispersion, are discussed. Furthermore, stability problems of such waves are investigated.