Multiple solutions for a nonlocal problem in Orlicz–Sobolev spaces

Using the mountain pass theorem combined with the minimum principle, we obtain a multiplicity result for a nonlocal problem in Orlicz–Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of spaces.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

An existence theorem of a positive solution to a semipositone Sturm–Liouville boundary value problem

We study a positive solution of the semipositone Sturm-Liouville boundary value problem in which the nonlinear term has no numerical lower bound. By considering the integration of certain limit growth functions and applying the Krasnosel’skii fixed point theorem on a cone, an existence theorem is proved, and a classical existence result is extended by this theorem.     

Neumann initial–boundary value problem for Benjamin–Ono equation

We consider the Neumann initial–boundary value problem for Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Initial boundary value problem for a coupled Camassa–Holm system with peakons

We investigate the homogeneous initial boundary value problem for a coupled Camassa–Holm system with peakons on the half line. We first establish the local well-posedness for the system. We then present a precise blowup scenario and several blowup results of strong solutions to the system. We finally give the blowup rate of strong solutions to the system when blowup occurs.     

Irregular boundary value problem for the Sturm–Liouville operator

We consider an eigenvalue problem for the Sturm–Liouville operator with nonclassical asymptotics of the spectrum. We prove that this problem, which has a complete system of root functions, is not almost regular (Stone-regular) but its Green function has a polynomial order of growth in the spectral parameter.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p$p$-laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.     

Mixed initial–boundary value problem for the Benjamin–Ono equation

We consider the mixed initial–boundary value problem for the Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals

Unbounded upper and lower solutions theories are established for the Sturm–Liouville boundary value problem of a second order ordinary differential equation on infinite intervals. By using such techniques and the Schäuder fixed point theorem, the existence of solutions as well as the positive ones is obtained. Nagumo conditions play an important role in the nonlinear term involved in the first-order derivatives.     

Inhomogeneous boundary value problem for nonstationary compressible Navier–Stokes equations

The existence of global weak renormalized solutions to the evolution flow problems for compressible Navier–Stokes equations is established. The in/out flow problem in a bounded domain in three spatial dimensions is considered. A general mathematical theory for the flow problem is developed. Bibliography: 15 titles.     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

An application of Levy-Caccioppoli’s global inversion theorem to a boundary value problem for the Soret effect

The Levy-Caccioppoli’s global inversion theorem is used to prove existence and uniqueness for a problem of heat and mass transfer. The relevant boundary value problem is first transformed in a suitable two-point problem for a first order differential equation.     

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.