Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

一类非线性Dirac方程组

本文寻求方程组 i sum from j=0 to 3 r~j _jΨ+L(Ψ■)Ψ=0的某些含两个零分量的解.     

Existence for the thermoelastic thermistor problem

We study a new model for the thermistor problem that consists of a system of dynamic thermoelasticity equations of displacement and a stationary charge conservation equation of electrical current. The heat source generated by Joule heating is quadratic in the gradient of the electrical potential. This system is nonlinear and degenerate since the electrical conductivity is assumed to be temperature dependent and vanishes at large temperatures. We establish the existence of a very weak solution, the so-called “capacity solution” for the model. The proof is based on regularization and time retarding.     

Hypergeometric Solutions of Trigonometric KZ Equations Satisfy Dynamical Difference Equations

The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ∈h where h⊂g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced by V. Tarasov and the second author (2000, Internat. Math. Res. Notices15, 801-829). We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to sl_N also satisfy the dynamical difference equations.     

Weak Solutions for the Vlasov-Poisson Initial-Boundary Value Problem with Bounded Electric Field

The aim of this work is to construct weak solutions for the three dimensional Vlasov-Poisson initial-boundary value problem with bounded electric field.The main in- gredient consists of estimating the change in momentum along characteristics of regular electric fields inside bounded spatial domains.As direct consequences,the propagation of the momentum moments and the existence of weak solution satisfying the balance of total energy are obtained.     

Convergence of periodically forced rank-type equations

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves, this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit, which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.     

A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this note, for the case of , we prove the existence of global-in-time finite energy weak solution of the equations of a two-dimensional magnetohydrodynamics with Coulomb force, where γ denotes the adiabatic exponent. The value is the optimal lower bound of γ to establish global-in-time finite energy weak solution under current frame.     

Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition

This paper deals with the blow-up of positive solutions for a nonlinear reaction-diffusion equation subject to nonlinear boundary conditions. We obtain the conditions under which the solutions may exist globally or blow up in finite time. Moreover, an upper bound of the blow-up time, an upper estimate of the blow-up rate, and an upper estimate of the global solutions are given. At last we give two examples to which the theorems obtained in the paper may be applied.     

三维不可压磁流体方程组的显式爆破解

该文构造了三维磁流体方程组的若干分离变量型和自相似型显式爆破解.     

On the definition of viscosity solutions for parabolic equations

In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.

     

The effect of domain shape on the number of positive and nodal solutions for semilinear elliptic equations

In this paper, we study the effect of domain shape on the number of positive and nodal (sign-changing) solutions for a class of semilinear elliptic equations. We prove a semilinear elliptic equation in a domain Ω$\Omega$ that contains m$m$ disjoint large enough balls has m²$m^2$ 2-nodal solutions and m$m$ positive solutions.     

New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham-Broer-Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.     

Isolated singularities for some types of curvature equations

We consider the removability of isolated singularities for the curvature equations of the form H_k[u]=0, which is determined by the kth elementary symmetric function, in an n-dimensional domain. We prove that, for 1?k?n−1, isolated singularities of any viscosity solutions to the curvature equations are always removable, provided the solution can be extended continuously at the singularities. We also consider the class of “generalized solutions” and prove the removability of isolated singularities.     

Using restrictions to accept or reject solutions of radical equations

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equation. We propose alternative solution procedures that are rigorous and simple to execute where the extraneous solutions can be identified without verification against the original equation. In this article, we review previous literature, establish and illustrate rigorous solution procedures for radical equations of depth 1 (i.e. equations where all radicals can be eliminated in one step), and deal with an ambiguity concerning the definition of real-valued solutions to radical equations. An application to defining the inverse function, resulting in a parametric radical equation, is also explained.     

Analytic general solutions of nonlinear difference equations

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of the following nonlinear second order difference equation,

Alternative solution of the inhomogeneous linear differential equation of order two

In this paper, we present an alternative method for solving the general inhomogeneous linear ordinary differential equation (ODE) of order two. The solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand. The main advantage of the method exposed herein is that the particular solution is computable from two different integrals. This allows the problem solver to choose the simplest integral with which to work with in order to get the final solution. For illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous Klein-Gordon equation.