Evolution equations with white-noise boundary conditions

The paper is devoted to nonlinear evolution equations with nonhomogenous boundary conditions of white noise type. Necessary and sufficient conditions for the existence of solutions in the linear case are given. It is also shown that if the nonlinearity satisfies appropriate dissipativity conditions the nonlinear equation has a solution as well. The results are applied to equations with polynomial nonlinearities     

On an evolution equation with acoustic boundary conditions

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota‐Goldstein work in the sense of considering a weaker internal damping and one more quadratic nonlinearity in the elastic string equation. Copyright © 2011 John Wiley & Sons, Ltd.     

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

In this paper, we investigate the correct solvability for the Laplace equation with a nonlocal boundary condition in the unit ball. The considered boundary operator is of fractional order. This problem is a generalization of the well‐known Bitsadze–Samarskii problem. Copyright © 2015 John Wiley & Sons, Ltd.     

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 a multi-point discrete boundary value problem

We study the existence and non-existence of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions. The proof of the existence of positive solutions is based upon the Schauder fixed point theorem.     

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.     

Diffusive logistic equation with non-linear boundary conditions

We analyze the solutions of a population model with diffusion and logistic growth. In particular, we focus our study on a population living in a patch, Ω⊆Rn with n?1, that satisfies a certain non-linear boundary condition and on its survival when constant yield harvesting is introduced. We establish our existence results by the method of sub-super solutions.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

On the free boundary problem to the two viscous immiscible fluids

In this paper, we prove the local solvability of the free boundary problem describing the motion of two layers of immiscible, heavy, viscous, incompressible fluid lying above an infinite rigid bottom and with surface tension on the interfaces, and global solvability near the equilibrium state.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Complete second order differential equations in Banach spaces with dynamic boundary conditions

In this paper, we deal with the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions. General wellposedness theorems are obtained (for the first time), which are used to solve the corresponding inhomogeneous problems. Examples of applications to initial boundary value problems for partial differential equations are also presented.     

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Ω₀ which is interior to the physical domain Ω⊂Rn. We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Ω₀ and converges uniformly to a continuous and positive function in .     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016     

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.     

On critical exponents for semilinear heat equations with nonlinear boundary conditions

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Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation − Δu = λ u − u ^p in Ω, with the nonlinear boundary condition ∂u/∂ν = u ^r on ∂Ω. Here Ω is a smooth bounded domain of with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.      

Solvability of the boundary value problem for stationary magnetohydrodynamic equations under mixed boundary conditions for the magnetic field

The global solvability of the boundary value problem for stationary magnetohydrodynamic equations under the Dirichlet boundary condition for the velocity and mixed boundary conditions for the magnetic field is proved.     

Fourth‐order compact schemes of a heat conduction problem with Neumann boundary conditions

In this article, a set of fourth‐order compact finite difference schemes is developed to solve a heat conduction problem with Neumann boundary conditions. It is derived through the compact difference schemes at all interior points, and the combined compact difference schemes at the boundary points. This set of schemes is proved to be globally solvable and unconditionally stable. Numerical examples are provided to verify the accuracy.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Boundary integral equations for bending of thermoelastic plates with transmission boundary conditions

The solution of an initial‐boundary value problem for bending of a piecewise‐homogeneous thermoelastic plate with transverse shear deformation is represented as various combinations of single‐layer and double‐layer time‐dependent potentials. The unique solvability of the boundary integral equations generated by these representations is proved in spaces of distributions. Copyright © 2009 John Wiley & Sons, Ltd.