Necat Polat

A analytic and numerical solution to a modified Kawahara equation and a convergence analysis of the method

Abstract In this paper, we present an Adomian’s decomposition method (shortly ADM) for numerical approximation traveling wave solutions of the modified Kawahara equation. The numerical solutions are compared with the known analytical solutions. We also prove the convergence of Adomian decomposition method (ADM) applied to the modified Kawahara equation.

Existence of global solutions for a multidimensional Boussinesq-type equation with supercritical initial energy

In this work, global weak solutions of the multidimensional Boussinesq-type equation with power type nonlinearity γ|u|p,γ>0 and supercritical initial energy is given by potential well method. Classical energy methods can not guarantee the global existence for this type of nonlinearity. As is known the functional I(u) defined for potential well method includes only the initial displacement, and by use of sign invariance of this functional one can only prove the global existence for critical and subcritical initial energy. In the case of supercritical initial energy such a functional fails to prove the global existence. A new functional is defined, which contains not only initial displacement, but also initial velocity.

Asymptotic behavior of a solution of the Cauchy problem for the generalized damped multidimensional Boussinesq equation

Abstract This work studies the Cauchy problem for the generalized damped multidimensional Boussinesq equation. By using a multiplier method, it is proven that the global solution of the problem decays to zero exponentially as the time approaches infinity, under a very simple and mild assumption regarding the nonlinear term.

On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity 𝛾|𝑢|𝑝 and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

Exponential decay and blow up of a solution for a system of nonlinear higher-order wave equations

This work studies a initial-boundary value problem of the weak damped nonlinear higher-order wave equations. Under suitable conditions on the initial datum, we prove that the solution decays exponentially and blows up with negative initial energy.

Blow-up phenomena and stability of solitary waves for a generalized Dullin-Gottwald-Holm equation

In this work, we consider the Cauchy problem of the generalized Dullin-Gottwald-Holm equation. We establish a blow-up result for the generalized Dullin-Gottwald-Holm equation. In addition to this, we investigate the stability of solitary wave solutions of the equation.

Blow up of a solution for a system of nonlinear higher-orderwave equations with strong damping terms

This work studies a initial-boundary value problem of the strong damped nonlinear higher-order wave equations. Under suitable conditions on the initial datum, we prove the blow up of the solution.

Blow up of Solution for the Generalized Boussinesq Equation with Damping Term

We consider the blow up of solution to the initial boundary value problem for the generalized Boussinesq equation with damping term. Under some assumptions we prove that the solution with negative initial energy blows up in finite time

Blow up of positive initial-energy solutions for a coupled nonlinear higher-order hyperbolic equations

This work studies an initial-boundary value problem of the coupled nonlinear higher-order hyperbolic equations with damping and source terms. Under suitable conditions on the initial datum, we prove the blow up of solutions with positive initial energy. We generalize some earlier results concerning the system.

Existence results for a nonlinear Timoshenko equation with high initial energy

The aim of the present paper is to study the initial-boundary value problem for a nonlinear Timoshenko equation with high energy initial data. Existence of global weak solutions is proved by sign preserving property of a new functional which is introduced for the potential well method.