On the solutions of a boundary value problem of linear thermoelasticity system with nonlocal conditions

This paper studies a boundary value problem with nonlocal conditions for a coupled system of linear thermoelasticity in one-dimensional case. Using an a priori estimate, we prove the uniqueness of the solution. Also, some explicit solutions are obtained by using the separation method.     

A nonlocal nonlinear diffusion equation with blowing up boundary conditions

We deal with boundary value problems (prescribing Dirichlet or Neumann boundary conditions) for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation. First, we prove existence, uniqueness and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. An illustrative example is also presented.     

Controllability of semilinear boundary problems with nonlocal initial conditions

The main purpose of this paper is the existence of solutions and controllability for semilinear boundary problems with nonlocal initial conditions. We show that the solutions are given by a variation of constants formula which allows us to study the exact controllability for this kind of problems with control and nonlinear terms at the boundary. The included application to a size structured population equation provides a motivation for abstract results.     

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of -periodic weak solutions under some reasonable assumptions.     

Global blow-up for a localized nonlinear parabolic equation with a nonlocal boundary condition

This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.     

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

Control and synchronizing nonlinear systems with delay based on a special matrix structure

This work presents a direct approach to design stabilizing controller for nonlinear systems with delay based on a special matrix structure and proves the validity of the approach according to Lyapunov–Krasovskii stable theorem and Linear Matrix Inequality—LMI. Control Lorenz system and synchronizing Rössler system with delay are taken as examples to explain the approach. Numerical simulations confirm the effectiveness of the approach proposed.     

A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions

Let p∈(1,N), Ω⊂RN a bounded W1,p-extension domain and let μ be an upper d-Ahlfors measure on ∂Ω with d∈(N−p,N). We show in the first part that for every p∈[2N/(N+2),N)∩(1,N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal Robin boundary conditions generates a (nonlinear) strongly continuous submarkovian semigroup on L²(Ω), and hence, the associated first order Cauchy problem is well posed on Lq(Ω) for every q∈[1,∞). In the second part we investigate existence, uniqueness and regularity of weak solutions to the associated quasi-linear elliptic equation. More precisely, global a priori estimates of weak solutions are obtained.     

Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition

This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.     

A comparison principle for nonlocal coupled systems of fully nonlinear parabolic equations

In this note, we establish a quite general comparison principle for a class of coupled systems of fully nonlinear parabolic equations subject to nonlocal boundary conditions.     

Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions

This paper deals with the oscillation problems of delay hyperbolic systems with impulses. Some sufficient conditions for oscillations of impulsive delay hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations

Inspired by the penalization of the domain approach of Lions and Sznitman, we give a sense to Neumann and oblique derivatives boundary value problems for nonlocal, possibly degenerate elliptic equations. Two different cases are considered: (i) homogeneous Neumann boundary conditions in convex, possibly non-smooth and unbounded domains, and (ii) general oblique derivatives boundary conditions in smooth, bounded, and possibly non-convex domains. In each case we give appropriate definitions of viscosity solutions and prove uniqueness of solutions of the corresponding boundary value problems. We prove that these boundary value problems arise in the penalization of the domain limit from whole space problems and obtain as a corollary the existence of solutions of these problems.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.