Some Notes on Reaction Diffusion Systems with Nonlinear Boundary Conditions

This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions.For the heat diffusion case,the necessary and sufficient conditions on the global existence of all positive solutions are obtained.For the general fast diffusion case,we get some conditions on the global existence and nonexistence of positive solutions.The results of this paper fill the some gaps which were left in this field.     

Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. It turns out that the existence or not of such solutions depends on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter, we prove the existence of solutions to this boundary value problem. For large values of the same parameter, we prove the nonexistence of solutions. We also provide rigorous bounds for the values of this parameter, which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions applied to an associated two‐point boundary value problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Almost automorphic solutions for abstract functional differential equations

For abstract linear functional differential equations with an almost automorphic forcing term, we establish a result on the existence of almost automorphic solutions, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

Multiplicity for Self-Dual Condensate Solutions in the Maxwell-Chern-Simons O(3) Sigma Model

In this paper, we study multiple existence of solutions of an elliptic system on a flat torus arising from the self-dual Maxwell-Chern-Simons O(3) sigma model. Using the Leray-Schauder degree, we find an explicit range of κ and q for which we have at least two vortex-antivortex solutions. For vortex solutions, we provide a complete picture of the range of κ and q for multiple existence of solutions.     

Strong solutions of periodic parabolic problems with discontinuous nonlinearities

We study the problem of finding time-periodic solutions of a parabolic equation with the homogeneous Dirichlet boundary condition and with a discontinuous nonlinearity. We assume that the nonlinearity is equal to the difference of two superpositionally measurable functions nondecreasing with respect to the state variable. For such a problem, we prove the principle of lower and upper solutions for the existence of strong solutions without additional constraints on the “jumping-up” discontinuities in the nonlinearity. We obtain existence theorems for strong solutions of this class of problems, including theorems on the existence of two nontrivial solutions.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.     

A critical elliptic problem for polyharmonic operators

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existence of solutions in domains perforated with the small holes. Some unstable solutions are obtained at higher level sets by Coron's topological method, provided that the minimizing solution does not exist.     

Second Darboux problem for the wave equation with a power-law nonlinearity

For the one-dimensional wave equation with a power-law nonlinearity, we consider the second Darboux problem and study the existence and uniqueness of a global solution, the existence of local solutions, and the absence of global solutions.     

PERIODIC SOLUTIONS AND ALMOST PERIODIC SOLUTIONS OF A NONAUTONOMOUS PREDATOR-PREY SYSTEM WITH STAGE-STRUCTURE AND DELAY

In this paper, by using the comparing theorem, Razumikhin-type theorem and V-function method, we consider a nonautonomous predator-prey system with stage-structure and time-delay. We get the sufficient conditions for the uniform persistence and the solutions global attractivity of this system. For a periodic system, we obtain the existence and uniqueness of a positive periodic solution of this system. For an almost periodic system, we prove the existence and the uniform asymptotic stability of the almost periodic solutions of this system.     

Regular and self-similar solutions of nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equations with nonlinear term |u|^ou. For some admissible α we show the existence of global solutions and we calculate the regularity of those solutions. Also we give some necessary conditions and some sufficient conditions on initial data for the existence of self-similar solutions.     

Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

In this paper, we study a non-local coupled system arising in the modeling of the dynamics of dislocation densities in crystals. For this system, the global existence and uniqueness are available only for continuous viscosity solutions. In the present paper, we investigate the global time existence of this system by considering BV initial data. Based on a fundamental uniform BV estimate and the finite speed of propagation property of this system, we show, in a particular setting, the global existence of discontinuous viscosity solutions of this problem.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

Existence and asymptotic behavior for a convection problem

We prove global existence and exponential decay of solutions for a system which arise in thermal convection flow. For sufficiently small initial data, these results improve previous ones in (Funkcial. Ekvac. 34 (1991) 449). Further, we investigate the behavior of solutions for arbitrarily large initial data. In particular, we show that the length of the interval on which we have existence and exponential decay is inverse proportional to the size of the initial data.     

Stability of a von Karman equation with infinite memory

In this paper, we consider a von Karman equation with infinite memory. For von Karman equations with finite memory, there is a lot of literature concerning on existence of the solutions, decay of the energy, and existence of the attractors. However, there are few results on existence and energy decay rate of the solutions for von Karman equations with infinite memory. The main goal of the present paper is to generalize previous results by treating infinite history instead of finite history.     

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.     

Existence theorems for solutions to random fuzzy differential equations

We examine a random fuzzy initial value problem with two kinds of fuzzy derivatives. For both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations. The existence of global solutions is also obtained.     

一类趋化性生物模型行波解的存在性

我们研究了一类趋化性(Chemotaxis)生物模型在不同情形下行波解的存在性.对D=0的一些情形,利用相平面分析的方法得到了行波解存在的充分必要条件;对D>0的一些情形得到了行波解的存在性,改进了Nagai和Ikeda原有的结果.     

Global solutions and invariant tori of difference-differential equations

We prove the existence of an m-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.     

The estimate for number of zeros of solutions of second order functional differential equations

In this paper, by extending the maximum principle, we study the number of zeros of solutions of second order functional differential equations. We obtain a sufficient condition for the existence of at most one zero of solutions on an interval. On this basis, we estimate the maximal number of zeros of solutions on a large interval. For illustrating the theoretical analysis, we also give two numerical simulation examples.