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 →∞.     

Analysis of a Free Boundary Problem Modeling Multi- Layer Tumor Growth in Presence of Inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little Holder spaces. Next we investigate asymptotic behavior of the solution. By making delicate analysis of spectrum of the linearization of the stationary free boundary problem and using the linearized stability theorem, we prove that if the surface tension coefficient γ is larger than γ^＊ 〉 0 the fiat stationary solution is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficient small.     

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.     

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.     

肿瘤生长的自由边界问题

本文介绍肿瘤生长的自由边界问题这一新兴研究方向的研究内容和进展状况.文章首先介绍肿瘤生长的数学建模历史、最新进展和一些重要的肿瘤生长模型,这些模型的数学形式是偏微分方程的自由边界问题.之后介绍近几年人们对这些自由边界问题所做严谨数学理论分析获得的一些主要成果,并简单介绍了证明这些成果用到的数学理论和方法.     

Bifurcation Analysis for a Free Boundary Problem Mo deling Growth of Solid Tumor with Inhibitors

This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.     

Free Boundary Value Problem for the Cylindrically Symmetric Compressible Navier-Stokes Equations with a Constant Exterior Pressure

This paper is concerned with the free boundary value problem (FBVP) for the cylindrically symmetric barotropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, the unique cylindrically symmetric strong solution is shown to exist globally in time and tend to a non-vacuum equilibrium state exponentially as time tends to infinity.     

On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \[ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.     

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.     

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.     

Uniqueness in a Free Boundary Problem

We study overdetermined boundary conditions for positive solutions to the p Laplacian in a bounded domain D. We show these conditions imply uniqueness in certain free boundary problems.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

On One Nonlocal Problem with Free Boundary

We investigate group-theoretic properties of a nonlocal problem with free boundary for a degenerating quasilinear parabolic equation. We establish conditions for the invariant solvability of this problem, perform its reduction, and obtain an exact self-similar solution.     

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.     

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

The Spin-Coating Process: Analysis of the Free Boundary Value Problem

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.     

趋化模型中的一个自由边界问题

本文就一个常见的生物趋化问题建立了相应的自由边界模型,对上述模型得到了其局部弱解的存在性.