GLOBAL ATTRACTOR OF A CLASS OF STRONGLY DAMPED NONLINEAR WAVE EQUATIONS

A class of strongly damped nonlinear wave equations are studied by using the technique of the operator decomposition,and the existence of the global compact attractor in space D(A)×V is obtained.

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R^2

In this article, we investigate the initial value problem（IVP） associated with the defocusing nonlinear wave equation on R^2 as follows：{δttu-△u=-u^3 u（0,x）=u0（x）,δtu＊（0,x）=u1（x,）where the initial data （uo,ul）∈H^s-1（R^2）It is shown that the IVP is global well-posedness in H^s（R^2）×H^s-1×H^s-1（R^2）for any 1 〉 s 〉2/5.The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].

Initial value problem for a class of fourth-order nonlinear wave equations

In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.

On the Cauchy Problem for a Class of Nonlinear Wave Equations with Damping Term

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the abovementioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.

A LOCAL ATTRACTOR FOR A NONLINEAR WAVE EQUATION ARISING IN ELASTIC WAVEGUIDE MODEL

This paper studies the asymptotic behavior of solutions to the initial boundary value problem for a nonlinear wave equation arising in an elastic waveguide model. It proves that under rather mild conditions on g the related solution semigroup possesses a local attractor.

The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.

The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term

We investigate the global well-posedness and the global attractors of the solutions for the Higher-order Kirchhoff-type wave equation with nonlinear strongly damping: . For strong nonlinear damping σ and ?, we make assumptions (H1) - (H4). Under of the proper assume, the main results are existence and uniqueness of the solution in proved by Galerkin method, and deal with the global attractors.

GLOBAL EXISTENCE AND L^p ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL SPACE

In this article, the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions. The author shows that a classical solution to the Cauchy problem exists globally in time under smallness condition on the initial perturbation. Furthermore, the author obtains the L^p （2 ≤ p ≤ ∞） decay estimates of the solution.

Regularity of Global Attractors for the Kirchhoff Wave Equation

In this paper, we mainly use operator decomposition technique to prove the global attractors which in ?for the Kirchhoff wave equation with strong damping and critical nonlinearities, are also bounded in .

Global Solution for Quasilinear Wave Equations with Viscosity and Nonlinear Perturbation

In this paper we prove the global existence and decay of solutions to the initial-boundary value problem for the quasilinear wave equations with viscosity and a nonlinear perturbation.

The Global Attractors and Dimensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation with Strong Damping

The initial boundary value problems for a class of high order Kirchhoff type equations with nonlinear strongly damped terms are considered. We establish the existence and uniqueness of the global solution of the problem by using prior estimates and Galerkin’s method under proper assumptions for the rigid term. Then the compact method is used to prove the existence of a compact family of global attractors in the solution semigroup generated by the problem. Finally, the Frechet differentiability of the operator semigroup and the decay of the volume element of linearization problem are proved, and the Hausdorff dimension and Fractal dimension of the family of global attractors are obtained.

Exponential attractors of the nonlinear wave equations

The (E, E)-type exponential attractors for the nonlinear wave equation were obtained by using the decomposition of operators and finite covering method.This results solves the open problem of Eden et al.

The Existence and Non-Existence of Global Solutions for a Nonlinear Wave Equation

This paper deals with the existence and uniqueness of the global solution of the initial boundary value problem of a class of wave equation.In the meantime,it gives the sufficient conditions of blow-up of the solution for the problem in finite time.

ATTRACTOR AND DIMENSION FOR STRONGLY DAMPED NONLINEAR WAVE EQUATION

In this paper, we prove the existence of the global attractor and obtain an estimate of the upper bound of Hausdorff dimension of the attractor for strongly damped nonlinear wave equation with Dirichlet boundary condition by introducing a new norm. The Hausdorff dimension obtained remains small for large damping, which conforms to the physical intuition.

THE INITIAL BOUNDARY VALUE PROBLEM FOR A CLASS OF NONLINEAR WAVE EQUATIONS WITH DAMPING TERM

The initial-boundary value problem for the four-order nonlinear wave equation with damping term is derived from diverse physical background such as the study of plate and beams and the study of interaction of water waves. The existence of the global weak solutions to this problem is proved by means of the potential well methods.

Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms

This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.

On the Strongly Damped Wave Equations with Critical Nonlinearities

We study the strongly damped wave equations with critical nonlinearities. By choosing suitable state spaces, we prove sectorial property of the operator matrix together with its adjoint operator, investigate the associated interpolation and extrapolation spaces, analysis the criticality of the nonlinearity with critical growth, and study the higher spatial regularity of the Y-regular solution by bootstrapping.

LARGE TIME BEHAVIOR OF SOLUTION TO NONLINEAR DIRAC EQUATION IN 1+1 DIMENSIONS

This paper studies the large time behavior of solution for a class of nonlinear massless Dirac equations in R^(1+1). It is shown that the solution will tend to travelling wave solution when time tends to infinity.

LOCAL ATTRACTORS FOR WEAKLY DAMPED FORCED KdV EQUATION IN THIN 2D DOMAINS

The existence of local attractors in thin 2D domains far the weakly damped forced KdV equation, whose principal operator is a non-self adjoint and non-sectorial one is given.