Periodic contrast structures in systems of the reaction-diffusion-advection type

In the present paper, we consider a nonlinear parabolic problem for an equation that is referred in applications to as the reaction-diffusion-advection equation and whose solutions have internal transition layers (contrast structures). For such equations, we construct the asymptotics of arbitrary-order accuracy and prove the existence and the Lyapunov stability.     

Non-local Problem for the Loaded Integral-differential Equation in Double-connected Domain

In this work an existence and uniqueness of solution of the non-local boundary value problemfor the loaded elliptic-hyperbolic type equation with integral-differential operations in double-connected domain have been investigated. The uniqueness of solution is proved by the method of integral energy using an extremum principle for the mixed type equations, and the existence is proved by the method of integral equations.     

Existence and uniqueness of solutions to an electrochemistry system

Existence and partial uniqueness results are established,under a variety of conditions, for a system of equations arising in electrochemistry problems. The main idea is the reduction of the system to a single equation involving nonlocal terms. Upper and lower solution procedures are then applied to show both existence and uniqueness. In this way, results are obtained in more than one space dimension.     

Boundary Value Problems for Odd Order Forward-Backward-Type Differential Equations with Two Time Variables

We study solvability of boundary value problems for odd order differential equations in time variables. The presence of a discontinuous alternating coefficient is a peculiarity of these equations. We prove existence and uniqueness theorems for the regular solutions of such an equation, i.e. those that have all Sobolev generalized derivatives entering the equation under study.     

关于结构弯扭屈曲理论的修正

本文讨论和指出了目前结构弯扭屈曲理论中所存在的问题．从而得出结论:目前的理论限制了结构的弯担变形过程,使得变形过程受制于线位移与转角位移按一定的先后顺序产生．为解决这一问题,提出了一种描述结构实际弯扭屈曲过程的新思路,从而导出了新的几何关系、建立了新的势能变分方程和新的中性平衡微分方程．实例说明了修正后的理论与原理论的不同．     

Standing Wave Solutions in Nonhomogeneous Delayed Synaptically Coupled Neuronal Networks

The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed systemof integral differential equations and a nonlinear scalar integral differential equation. It will be shown that there exist six standing wave solutions (u(x,t),w(x,t))=(U(x),W(x)) to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave solutions u(x,t)=U(x) to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.     

Feedback theory extended for proving generation of contraction semigroups

Recently, the following novel method for proving the existence of solutions for certain linear time-invariant PDEs was introduced: The operator associated with a given PDE is represented by a (larger) operator with an internal loop. If the larger operator (without the internal loop) generates a contraction semigroup, the internal loop is accretive, and some non-restrictive technical assumptions are fulfilled, then the original operator generates a contraction semigroup as well. Beginning with the undamped wave equation, this general idea can be applied to show that the heat equation and wave equations with damping are well-posed. In the present paper we show how this approach can benefit from feedback techniques and recent developments in well-posed systems theory, at the same time generalizing the previously known results. Among others, we show how well-posedness of degenerate parabolic equations can be proved.     

Weak solution for stochastic differential equations with terminal conditions

The notion of weak solution for stochastic differential equation with terminal conditions is introduced. By Girsanov transformation, the equivalence of existence of weak solutions for two-type equations is established. Several sufficient conditions for the existence of the weak solutions for stochastic differential equation with terminal conditions are obtained, and the solution existence condition for this type of equations is relaxed. Finally, an example is given to show that the result is an essential extension of the one under Lipschitz condition ong with respect to (Y,Z).     

A simplified quantum energy-transport model for semiconductors

The existence of global-in-time weak solutions to a quantum energy-transport model for semiconductors is proved. The equations are formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime. They consist of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; an elliptic nonlinear heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The existence proof is based on an entropy-type estimate, exponential variable transformations, and a fixed-point argument. Furthermore, we discretize the equations by central finite differences and present some numerical simulations of a one-dimensional ballistic diode.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Investigation of solutions to one family of mathematical models of living systems

We consider a family of integral equations used as models of some living systems. We prove that an integral equation is reducible to the equivalent Cauchy problem for a non-autonomous differential equation with point or distributed delay dependently on the choice of the survival function of system elements. We also study the issues of the existence, uniqueness, nonnegativity, and continuability of solutions. We describe all stationary solutions and obtain sufficient conditions for their asymptotic stability. We have found sufficient conditions for the existence of a limit of solutions on infinity and present an example of equations where the rate of generation of elements of living systems is described by a unimodal function (namely, the Hill function).     

On the theory of convolution equations of the third kind

The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived.     

On Analytical Results for the Optimal Control of the quasi-stationary Radiative Heat Transfer System

We state an optimal control problem of the coupled quasi-stationary radiative heat equations consisting of the radiative transfer equation and the instationary heat transfer equation that model radiative-conductive heat transfer. We give an existence and uniqueness result for the state equations and the adjoint equations of the quasi-stationary radiative heat transfer system. For the optimal control problem the existence of a minimizer is proven. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

Single dopant diffusion in semiconductor technology

The paper deals with the analysis of pair diffusion models in semiconductor technology. The underlying model contains reaction‐drift‐diffusion equations for the mobile point defects and dopant‐defect pairs as well as reaction equations for immobile dopants which are coupled with a non‐linear Poisson equation for the chemical potential of the electrons. For homogeneous structures we present an existence and uniqueness result for strong solutions. Starting with energy estimates we derive further a priori estimates such that fixed point arguments due to Leray–Schauder guarantee the solvability of the model equations. Copyright © 2004 John Wiley & Sons, Ltd.     

A diagonal splitting method for solving semidiscretized parabolic partial differential equations

In this work, a diagonal splitting idea is presented for solving linear systems of ordinary differential equations. The resulting methods are specially efficient for solving systems which have arisen from semidiscretization of parabolic partial differential equations (PDEs). Unconditional stability of methods for heat equation and advection–diffusion equation is shown in maximum norm. Generalization of the methods in higher dimensions is discussed. Some illustrative examples are presented to show efficiency of the new methods.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

各项异性椭圆方程基本解的存在性

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.     

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.