Search

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]

In this paper a Galerkin method is used to construct series solutions of a homogeneous Sturm-Liouville problem defined on [0, &pie;]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L² [0,&Pie;]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Cesaro-one summability method. Therefore, in the Cesaro-one sense, every continuous function / o n [0, &Pie;] is the uniform limit of solutions of non-homogeneous Sturm-Liouville problems. [ABSTRACT FROM AUTHOR]

We consider the large time asymptotic behavior of solutions to the initial-boundary value problem [This symbol cannot be presented in ASCII format]. We find large time asymptotic formulas of solutions for three different cases 1) a± = ±1, 2) a± = [This symbol cannot be presented in ASCII format]1 and 3) a± = 0. [ABSTRACT FROM AUTHOR]

The article examines the existence of solutions for weighted p(t)-Laplacian system multipoint boundary value problems. The existence of solutions is given via Leray-Schauder degree when the nonlinearity term f (t,.,.) satisfies general growth condition. The study of differential equations and variational problems with variable exponent growth conditions is an interesting topic as many results have been obtained on these problems. Banach spaces are used in this study to denote positive constants to satisfy the growth condition of the nonlinearity term f.

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

We discuss differential equations with nonlinear boundary conditions. We formulate sufficient conditions under which problems with deviating arguments have quasisolutions or solutions. To obtain the results, we apply the method of monotone iterations. [ABSTRACT FROM AUTHOR]

We consider the problem of solvability and optimization for a pseudohyperbolic operator of the general form. We prove theorems on existence and uniqueness for various right-hand sides of the equation. The results obtained are applied to the problem of trajectory-final controllability. [ABSTRACT FROM AUTHOR]