一个肿瘤生长自由边界问题解的渐近性态

该文研究根据Byrne和Chaplain的思想建立的一个描述抑制物作用下无坏死核肿瘤生长的数学模型, 这个模型是一个非线性反应扩散方程组的自由边界问题. 作者运用反应扩散方程理论中的上下解方法结合自由边界问题的迭代技巧, 研究了解的渐近性态, 在营养物消耗函数f、抑制物消耗函数g和肿瘤细胞繁衍函数S的一些一般条件下,证明当常数c₁,c₂(肿瘤细胞分裂速率和营养物、抑制物扩散速率的比值)都非常小时,在一定的初边值条件下肿瘤趋于消失,在另外一些初边值条件下肿瘤半径趋于一个常数,进而时变解将趋于一个稳态解.     

Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure

In this paper, a diffusive predator-prey model with nonlocal delay and stage structure is investigated. By using the cross iteration method and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. Numerical simulations are carried out to illustrate the theoretical results.     

Global Existence for a Parabolic-hyperbolic Free Boundary Problem Modelling Tumor Growth

In this paper we study a free boundary problem modelling tumor growth, proposed by A. Friedman in 2004. This free boundary problem involves a nonlinear second-order parabolic equation describing the diffusion of nutrient in the tumor, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells, respectively. By applying Lp theory of parabolic equations, the characteristic theory of hyperbolic equations, and the Banach fixed point theorem, we prove that this problem has a unique global classical solution.     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling wave solutions for a Lotka-Volterra reaction-diffusion competition model with time delay. By constructing upper and lower solutions, we obtain the precise minimum wave speed of traveling waves under certain conditions. Our results also extend the known results on the minimum wave speed for Lotka-Volterra competition model without delay.     

Uniqueness of traveling wave solutions for a biological reaction-diffusion equation

The existence of traveling wave solutions for a reaction-diffusion, which serves as models for microbial growth in a flow reactor and for mathematical epidemiology, was previously confirmed. However, the problem on the uniqueness of traveling wave solutions remains open. In this paper we give a complete proof of the uniqueness of traveling wave solutions.     

Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

一个肿瘤生长自由边界问题解的整体存在性和唯一性

该文研究一个反应扩散方程组的自由边界问题,它来源于描述抑制物作用下无坏死核肿瘤生长的数学模型.作者运用抛物型方程的L_p理论和压缩映照原理,证明了这个问题局部解的存在唯一性,然后用延拓方法得到了整体解的存在唯一性.     

Finite Travelling Waves for a Semilinear Degenerate Reaction-Diffusion System

In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system is studied, where 0 < α _i < 1, m _ij ≥ 0 and ∑ ^N _{j =1} m _ij > 0, i, j = 1,...,N. Necessary and sufficient conditions on existence and large time behaviours of FTWs of (I) are obtained by using the matrix theory, Schauder's fixed point theorem, and upper and lower solutions method. Received July 3, 1997, Revised June 23, 1999, Accepted March 29, 2000     

一类半线性退化反应扩散方程组的有限行波解

本文考虑一类半线性退化反应扩散方程组有限行波解的存在性与不存在性,以及有限行波解存在的充要条件与大时间行为被获得.     

Asymptotic Behaviour of Solutions of a Multidimensional Moving Boundary Problem Modeling Tumor Growth

We study a moving boundary problem modeling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumor's body, respectively. The driving mechanism of the evolution of the tumor surface is governed by Darcy's law. Finally surface tension effects on the moving boundary are taken into account which are considered to counterbalance the internal pressure. To put our analysis on a solid basis, we first state a local well-posedness result for general initial data. However, the main purpose of our study is the investigation of the asymptotic behaviour of solutions as time goes to infinity. As a result of a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in the C ^m+μ-norm with m ≥ 3 and μ ∈ (0,1), then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly shifted) ball, provided the surface tension coefficient γ is larger than a positive threshold value γ_*. In the case 0 < γ < γ_* the radially symmetric equilibrium is unstable.     

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.     

Analysis of a System Describing Proliferative-Quiescent Cell Dynamics

Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis. Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.     

二次谐波产生耦合方程组的解

本文讨论了非线性光学中的二次谐波产生的耦合方程组 ,利用变分法证明了耦合方程组非平凡解的存在性 ,然后进行了数值模拟 ,实验结果表明文中方法比经典的非线性光学中的方法有较大的改进 ,这对优化光倍频器件的设计将有所帮助 .     

Existence and Nonexistence of Traveling Wave Solutions for a Bistable Reaction-Diffusion Equation in an Infinite Cylinder Whose Diameter is Suddenly Increased

ABSTRACT

We consider a reaction-diffusion equation of bistable type in a square cylinder whose diameter varies with Neumann boundary conditions in dimension 2 and 3. We prove the nonexistence of generalized traveling wave solution of this equation when the diameter is suddenly strongly increased. At the same time, we prove that the solution of the equation with an exponentially decreasing initial condition cannot pass over a certain threshold far enough in the direction of propagation.

The proof is divided in two steps. First, we extend the solution in the cylinder to a solution of the same equation in the half space. Then we overestimate it using Green's functions.     

一个药物作用肿瘤生长自由边界问题整体解的存在唯一性

该文研究一个描述药物作用下肿瘤生长的数学模型,这个肿瘤模型是对Jackson模型的一个改进,其数学形式是由一个二阶非线性抛物型方程与两个一阶非线性偏微分方程组耦合而成的自由边界问题.通过运用抛物型方程的L~P理论与一阶偏微分方程的特征方法,并利用Banach不动点定理,证明了该问题存在唯一的整体经典解.     

带分布时滞的具有尺度结构的种群模型的异步指数增长

研究一个带分布时滞的具有尺度结构的种群模型.模型将个体分为"活跃"期和"休眠"期两个阶段研究.利用算子半群的理论证明了此模型的适定性并证明此模型的解具有异步指数增长的状态.     

一类具有时变系数梯度源项的弱耦合反应-扩散方程组解的爆破分析

该文研究了具有时变系数梯度项的弱耦合反应-扩散方程组齐次Dirichlet初边值问题解的爆破现象.结合修正微分不等式技巧及比较原理,得到了在若干个不同测度意义下解的整体存在性与有限时刻发生爆破的充分条件,并在高维空间中导出了爆破解的爆破时间界的估计.