Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.     

非线性电报方程组的3个双周期正解问题

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.     

Existence of solutions to differential inclusions in Banach spaces

I.IntroductionOwingtothefactthatthesystemwithwhicheconomics,biologyandsoonaredefinitelyconcernedtohaveuncertainty,includingthatcausedbyrestrictiononknowledge.asformathematicaltreatment,generallytheycanonlybedescribedbymeansofdifferentialinclusionsinsteadofdifferentialequations.Itiswell-knownthatthederivativex')ofasolutionx.)tothedifferentialequationx'(t)=f(t,x(t))inheritsthereqularityproperitiesofthemapfandthefunctionx(').Thisisnolongerthecasewithdifferentialinclusionsandisoneofthereasonswhyt…     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

Reduction de systemes differentiels perturbes

One considers a perturbed ordinary differential system which is then reduced to the non-perturbed corresponding system; i.e. the existence of a global relation between the solutions of the two systems is established. From this it is easy to deduce the difference between two correspondent solutions, and the theorems of existence for particular solutions (periodic, quasiperiodic,…) or integral manifolds, of the initial system.     

On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.     

Schauder estimates for parabolic equation of bi-harmonic type

Global Schauder estimates for the initial-value parabolic problem of the bi-harmonic type are proved,and the existence and uniqueness of the solutions in the suitable space are obtained.Similarly to the second-order case,first a formal expression of solutions by the Fourier transform is obtained,and then the regularity,uniqueness and existence of solutions using the potential theory and approximation argument are got. out approach is simple and straightforward.     

Single-Pulse Solutions for Oscillatory Coupling Functions in Neural Networks

We study the existence and linear stability of stationary pulse solutions of an integro-differential equation modeling the coarse-grained averaged activity of a single layer of interconnected neurons. The neuronal connections considered feature lateral oscillations with an exponential rate of decay and variable period. We identify regions in the parameter space where solutions exhibit areas of excitation with single- and dimpled-pulses. When the gain function reduces to the Heaviside function, we establish existence of single-pulse solutions analytically. For a more general gain function, we include numerical support of the existence of pulse-like solutions. We then study the linear stability behavior of these solutions.     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

On the periodic solutions of differential inclusions and applications

The periodic problem of evolution inclusion is studied and its results are used toestablish existence theorems of periodic solutions of a class of semi-linear differential inclusion.Also existence theorem of the extreme solutions and the strong relaxation theorem are given forthis class of semi-linear differential inclusion. An application to some feedback control systems isdiscussed.     

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Existence of Solutions for the Ericksen-Leslie System

We study the general Ericksen-Leslie system, which describes the flow of liquid crystal materials. The dissipation property of the system is established and is used to prove the global existence of weak solutions. We also study the existence of classical solutions and the asymptotic stability of the solutions. (Accepted: January 15, 2000)?Published online September 12, 2000     

Energy Dissipation and Regularity for a Coupled Navier–Stokes and Q-Tensor System

We study a complex non-Newtonian fluid that models the flowof nematic liquid crystals. The fluid is described by a system that couples a forced Navier–Stokes system with a parabolic-type system. We prove the existence of global weak solutions in dimensions two and three.We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use cancellations that allow its existence to prove higher global regularity in dimension two. We also show the weak–strong uniqueness in dimension two.     

挠性联结双体航天器的稳定性与分岔

研究圆轨道内受万有引力矩作用的挠性联结双体航天器在轨道平面内的姿态运动,讨论其相对轨道坐标系统平衡状态的稳定性与分岔。提出判平衡方程非平凡解存在性的几何方法,并应用Ｌｉａｐｕｎｏｖ直接法、Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化方法和奇异性理论导出解析形式的稳定性与分岔的充要条件,从而对系统的全局运动性态作出定性的描述。     

Existence of Traveling-Wave Solutions for Hyperbolic Systems of Balance Laws

This paper is concerned with traveling-wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a Kawashima-like condition. We are interested in the case where the traveling-wave equations have a singularity, which is absent for 2 × 2 systems satisfying the two conditions. To deal with the singularity, we reduce the problem to a parametrized one without singularity by using the center manifold theorem. For the parametrized problem, we prove the existence of solutions by modifying an existing argument in the literature. In this way, we show the existence of traveling-wave solutions.     

Solitons in plasma with a hall dispersion

A system of equations of perfect magnetohydrodynamics is considered with allowance for Hall currents. The study of one-dimensional steady solutions which are damped at infinity can be reduced to the investigation of a Hamiltonian dynamic system with right-hand sides that are not single valued. A qualitative investigation of the system is carried out, with the determination of the region of existence of the given solutions. The solutions have the form of solitary waves — solitons. An exact solution in quadratures is obtained, which describes the structure of the solitons. The existence of two solitons of the Alfvén type is indicated. The existence domain of the corresponding solutions is analyzed. In the limiting cases of magnetosonic and Alfvén solitons, the solutions are expressed in explicit form in elementary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–168, September–October, 1985.     

EXISTENCE OF BOUNDED SOLUTIONS ON THE REAL LINE FOR LIENARD SYSTEM

The existence of monotone and non-monotone solutions of boundary value problem on the real line for Lienard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Lienard system and the system given by symmetric transformation or quasi-symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi-stable, the existence of infinitely many monotone solusion is obtained.     

Homogeneous Cooling States¶are not always¶Good Approximations to Granular Flows

A widespread belief in the study of granular flow is the existence of “homogeneous cooling states”, i.e., self-similar solutions which would attract all solutions, faster than the equilibrium solution does. In most cases, the existence of these self-similar solutions is an open problem. Here we consider a one-dimensional model, which has been used for some years, and for which simple self-similar solutions do exist. However, we prove that the approximation is quite poor. Our proof makes use of the powerful and simple tools of mass transportation, and exploits the structure of the evolution equation, seen as a nonlinear transport equation.     

Moore-Gibson-Thompson theory for thermoelastic dielectrics

We consider the system of equations determining the linear thermoelastic deformations of dielectrics within the recently called Moore-Gibson-Thompson(MGT)theory.First,we obtain the system of equations for such a case.Second,we consider the case of a rigid solid and show the existence and the exponential decay of solutions.Third,we consider the thermoelastic case and obtain the existence and the stability of the solutions.Exponential decay of solutions in the one-dimensional case is also recalled.