Existence,semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model

The existence, semiclassical limit and long-time behavior of weak solutions to the transient quantum drift-diffusion model are studied. Using semi-discretization in time and entropy estimate, we get the global existence and semiclassical limit of nonnegative weak solutions to one-dimensional isentropic model with nonnegative initial and homogeneous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, we obtain an inequality of the periodic weak solution to this model (or its isothermal case) which shows that the solution exponentially approaches its mean value as time increases to infinity.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Existence and attractivity of time periodic solutions for Nicholson's blowflies model with nonlinear diffusion

This paper is concerned with the existence and attractivity of time periodic solutions for one‐dimensional Nicholson's blowflies model with nonlinear diffusion. By constructing some suitable Lyapunov functionals and combining with Leray–Schauder fixed point theorem, we establish the existence of nonnegative time periodic solutions. Using the method of upper and lower solutions and its associated monotone iterations, we obtain the existence of the attractor for this model. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of time periodic solutions for the Nicholson's blowflies model with Newtonian diffusion

This paper is concerned with the existence of periodic solutions of the Nicholson's blowflies model with Newtonian diffusion. By constructing some suitable Lyapunov functionals and combining with Leray–Schauder fixed point theorem, we establish the existence of nonnegative time periodic solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms

In this paper, the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of Leray-Schauder's degree.     

On an unstable thin-film equation in multi-dimensional domains

We study the existence of weak and strong solutions to the initial-boundary value problem for a thin-film type equation with unstable diffusion in multi-dimensional domains. Depending on the initial data and the parameter values, we prove local and global in time existence of nonnegative weak and strong solutions.     

Global positive periodic solutions of age-dependent competing systems

This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn’s asymptotic fixed point theorem.     

A multidimensional nonlinear sixth-order quantum diffusion equation

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.     

On the existence of periodic solutions in the thermistor problem with degenerate thermal conductivity

Abstract This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65     

Existence of Physical Solutions to the Boundary Value Problem of an Equation in Modeling Oil Film Spreading Over a Solid Surface

A boundary vaJue problem to a fourth order nonlinear degeneraw para-holic equation arising in modeling oil film spreading over a solid surface is studied in the present paper. Basing on the consideration of physical phenomenon describrd by the original model, the authors forcus mainly on the case of two-dimensional space. The existence of nonnegative and radial solutions is established for nonnegative initial datum.     

BD entropy and Bernis–Friedman entropy

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis–Friedman (called BF) dissipative entropy introduced to study the lubrication equations. Different dissipative entropies are obtained playing with the drag terms on the viscous shallow-water equations. It helps for instance to prove the global existence of nonnegative weak solutions to the lubrication equations starting from the global existence of nonnegative weak solutions to appropriate viscous shallow-water equations.     

Existence of Non-trivial Nonnegative Periodic Solutions for a Nonlinear Diffusion System

In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.     

一类非线性扩散系统非平凡非负周期解的存在性

In this paper,by the method of upper and lower solutions,we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.     

Periodic solutions for a class of degenerate parabolic equations with Neumann boundary conditions

In this paper, we investigate a class of degenerate parabolic periodic equations with nonlocal terms under Neumann boundary conditions. By using the theory of the Leray–Schauder degree, we obtain the existence of non-trivial nonnegative periodic solutions.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

Nonnegative solutions to reaction–diffusion system with cross-diffusion and nonstandard growth conditions

We establish the existence of nonnegative weak solutions to nonlinear reaction–diffusion system with cross-diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.     

A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease

We provide existence results for nonnegative solutions to a class of reaction-diffusion systems with cross-diffusion modeling the spread of an epidemic disease. The existence of weak solutions is proved by means of an approximation process, the Faedo-Galerkin method, and a compactness argument. Under additional assumptions a global existence result for classical solutions is derived upon using interpolation results between Banach spaces.     

一个具有可测数据的化学反应扩散系统

该文证明一个化学反应扩散系统当初始数据是有界非负犚犪犱狅狀测度时弱解整体存在,且给出了解收敛于正平衡解的充要条件．     

一类多维非线性渗流方程解的有限传播性质

本文讨论具有强非线性源与对流项的渗流方程,利用其解的Harnack不等式得到弱解的具有限传播速度的性质或正解分界面的存在性与增长性.     

Periodic behaviour for the evolutionary dam problem and related free boundary problems Evolutionary dam problem

The time dependent dam problem describing the seepage of a compressible or incompressible fluid in a porous dam is studied. We prove existence of solutions in a suitable weak sense, and uniqueness for rectangular dams. Existence or periodic solutions is established and questions ofstability and periodic behavior for large time are studied.