The Existence of Global Solutions to a Fluid Dynamic Model for Materials for Korteweg Type

We discuss the existence of global solutions to a fluid dynamic model for materials of Korteweg type, and discuss the three dimensional nonisothermal motion. The higher order derivatives of density in the constitutive relation makes it necessary to introduce an interstitial working term in the energy equation. This makes the energy estimate more involved. Nevertheless, it is possible to show the existence of global solutions by the euergy method.     

一类具交错扩散的强耦合退化抛物方程组解的整体存在与非存在性

本文研究了一类具交错扩散的强耦合拟线性退化抛物方程组初边值问题正古典解的局部存在,整体存在与非整体存在性.利用正则化方法和先验估计技巧证明了该问题正古典解的局部存在性,并且分别给出了该问题是否存在整体古典解的充分条件.结果表明当种群内竞争强于种群间互惠作用时,此问题存在整体解;而当两种群具有强互惠作用时,所有解都是非整体的.     

Dynamics of a delayed business cycle model with general investment function

The aim of this paper is to study the dynamics of a delayed business cycle model with general investment function. The model describes the interaction of the gross product and capital stock. Furthermore, the delay represents the time between the decision of investment and implementation. Firstly, we show that the model is well posed by proving the global existence and boundedness of solutions. Secondly, we determine the economic equilibrium of the model. By analyzing the characteristic equation, we investigate the stability of the economic equilibrium and the local existence of Hopf bifurcation. Also, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theory. Moreover, the global existence of bifurcating periodic solutions is established by using the global Hopf bifurcation theory. Finally, our theoretical results are illustrated with some numerical simulations.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

Coexistence and Asymptotic Periodicity in a Competition Model of Plankton Allelopathy

In this paper, the two species Lotka-Volterra competition model of plankton allelopathy from aquatic ecology is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions and self-diffusions are weak. The existence of the positive T-periodic solutions and the global stability as well as the global attractivity for the parabolic system are also given.     

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.     

Global smooth solutions to relativistic Euler-Poisson equations with repulsive force

In this paper, we are concerned with the global existence of smooth solutions for the one dimen- sional relativistic Euler-Poisson equations： Combining certain physical background, the relativistic Euler-Poisson model is derived mathematically. By using an invariant of Lax＇s method, we will give a sufficient condition for the existence of a global smooth solution to the one-dimensional Euler-Poisson equations with repulsive force.     

Global Weak Solutions of the Cauchy Problem to a Hydrodynamic Model for Semiconductors

In this paper we are concerned with the global existence of weak solutions of the Cauchy problem for a simplified one-dimensional hydrodynamic model for semiconductors. Convergence of approximate solutions derived by the fractional step Lax-Friedrichs scheme is established by using the compensated compactness method.     

Existence of positive periodic solutions of an SEIR model with periodic coefficients

An SEIR model with periodic coefficients in epidemiology is considered. The global existence of periodic solutions with strictly positive components for this model is established by using the method of coincidence degree. Furthermore, a sufficient condition for the global stability of this model is obtained. An example based on the transmission of respiratory syncytial virus (RSV) is included.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

On the stability of a population model with nonlocal dispersal

This paper is concerned with a nonlocal dispersal population model with spatial competition and aggregation. We establish the existence and uniqueness of positive solutions by the method of coupled upper-lower solutions. We obtain the global stability of the stationary solutions.     

Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model

We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller–Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller–Segel model with global existence of solutions.     

Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model

We study the global existence of weak solutions to a reduced gravity two-and-a-half layer model appearing in oceanic fluid dynamics in two-dimensional torus. Based on Faedo–Galerkin method and weak convergence method, we construct the global weak solutions which are renormalized in velocity variable, where the technique of renormalized solutions was introduced by Lacroix-Violet and Vasseur (2018). Besides, we prove that the renormalized solutions are weak solutions, which satisfy the basic energy inequality and Bresch–Desjardins entropy inequality, but not the Mellet–Vasseur type inequality. In the proof, we use the reduced gravity two-and-a-half layer model with drag forces and capillary term as approximate system. It should be pointed out that only when the capillary term vanishes, we prove the existence of renormalized solution to the approximation system, which is different from Lacroix-Violet and Vasseur (2018) with the quantum potential.     

Stochastic Asymptotic Stability of Nowak-May Model with Variable Diffusion Rates

We consider a stochastically perturbed Nowak-May model of virus dynamics within a host. We prove the global existence of unique strong solution. Using the Lyapunov method, we found sufficient conditions for the stochastic asymptotic stability of equilibrium solutions of this model.     

捕食者-食饵-互惠交错扩散模型解的整体存在性

本文应用Sobolev嵌入定理,能量估计和bootstrap技巧证明一类捕食者-食饵-互惠交错扩散模型在空间维数小于10时古典解的整体存在性.     

一类半线性双温度热传导方程整体强解的存在性

给出了一类半线性双温度热传导方程的初边值问题整体强解的存在条件,利用位势井方法证明了整体强解的存在性定理,且证明了方程的解或解对X的某些导数的L^2模估计式．     

On a coupled system of reaction–diffusion-transport equations arising from catalytic converter

This paper is concerned with two mathematical models which describe the transient behavior of a catalytic converter in automobile engineering. The first model consists of a coupled system of a heat-conduction equation and two integral equations while the second model involves only one integral equation. It is shown that for any nonnegative initial and boundary functions the three-equation model has a unique bounded global solution while the solution of the two-equation model blows up in finite time. The proof for the global existence and finite-time blow-up property of the solution is by the method of upper and lower solutions and its associated monotone iteration. This method can be used to develop computational algorithms for numerical solutions of the coupled systems.     

Global existence and blow-up of solutions for some hyperbolic systems with damping and source terms

This paper deals with the global existence and blow-up of solutions to some nonlinear hyperbolic systems with damping and source terms in a bounded domain. By using the potential well method, we obtain the global existence. Moreover, for the problem with linear damping terms, blow-up of solutions is considered and some estimates for the lifespan of solutions are given.     

一类泛函微分方程的周期解的存在性、唯一性及全局吸引性

该文研究了一类具有分布滞量的微分系统的周期解的存在性、唯一性及全局吸引性等问题.利用不动点方法和Lyapunov泛函方法,建立了保证该类系统周期解的存在性、唯一性、一致稳定性及全局吸引性的充分条件.     

Existence and Attractivity of <Emphasis Type="Italic">k</Emphasis>-Pseudo Almost Automorphic Sequence Solution of a Model of Bidirectional Neural Networks

In this paper we discuss the existence and global attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional cellular neural networks. We consider the corresponding difference equation analogue of the model system using suitable discretization method and obtain certain conditions for the existence of solution. The k-pseudo almost automorphic sequence solutions generalize the results of pseudo almost periodic, almost periodic and almost automorphic sequences solutions. Moreover the results proved in this paper are new and compliment the existing one.