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.     

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.     

On a Hele-Shaw Flow Problem with Free and Solid Boundary Components

The purpose of this work is to study a one phase Hele-Shaw fluid flow occupying a time variable domain \(\Omega (t)\), due to the injection of the fluid with a constant rate at a single point of the initial domain \(\Omega (0)\), and in the presence of a fixed solid body \(\Omega _0\). We show the short time existence and uniqueness of the solution for the corresponding boundary value problem in the three dimensional case and in the absence of surface tension.     

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.     

On Free Boundary Problem for the Non-Newtonian Shear Thickening Fluids

The aim of this paper is to explore the free boundary problem for the NonNewtonian shear thickening fluids. These fluids not only have vacuum, but also have strong nonlinear properties. In this paper, a class of approximate solutions is first constructed, and some uniform estimates are obtained for these approximate solutions. Finally, the existence of free boundary problem solutions is proved by these uniform estimates.     

On a Boundary Layer Problem

This is a continuation of our earlier article concerning the boundary-value problem     where A , B are prescribed constants, and 0 < ε ≪ 1 is a small positive parameter. In that article, we assumed the coefficients a ( x ) and b ( x ) are sufficiently smooth functions with the behavior given by a ( x ) ∼ αx and b ( x ) ∼ β as x → 0, where α > 0 and β / α ≠ 1, 2, 3,…. In the present article, we are concerned with the case α < 0 and β / α ≠ 0, −1, −2,…. An asymptotic solution is obtained for the problem, which holds uniformly for all x in [ x ₋, x ₊]. Our result is proved rigorously, and shows that a previous result in the literature is incorrect.     

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.     

一个反应扩散方程的自由边界问题

该文研究了一个描述原细胞生长的反应扩散方程的自由边界问题.利用非线性分析中的线性化思想和偏微分方程的估计理论,证明了该自由边界问题局部古典解的存在唯一性.     

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 →∞.     

扰动方法研究R~3中自由边值问题

该文运用扰动方法研究了在R~3中如下自由边界问题其中λ为常数,Ω为R~3中的单位球,证明了,在一定条件下,存在ε_00,只要0εε_0,上述问题的解存在以及该解在ε→0时的一些性质.     

Nonuniqueness in a Free Boundary Problem from Combustion

We study a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. Vázquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion. A. Petrosyan was supported in part by NSF grant DMS-0701015. N.K. Yip was supported in part by NSF grant DMS-0406033.     

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

The Well Posedness in a Parabolic Multiple Free Boundary Problem

We consider a free boundary problem in a parabolic partial differential equation with multiple interfacial curves which is reduced to a reaction-diffusion equation. The forcing term of this problem is not continuously differentiable and thus we use Green's function to make a regular one. The existence, uniqueness and dependence. on initial conditions will be shown in this paper.     

On a Moving Boundary Problem from Biomechanics

A moving boundary problem arising in biomechanical diffusiontheory which has previously been investigated by Crank &Gupta (1972a, b) is studied using a different method of solution.The method is based on an integral equation for the functiondefining the position of the moving boundary and an integralformula for the concentration. The integral equation is solvedasymptotically for small times and numerically during the entiremotion of the boundary. The concentration is estimated asymptoticallyfor small times and computed by numerical quadrature at laterinstants. The results are compared with those of Crank &Gupta. In most cases the agreement is fair.     

Numerical Solution of a Parabolic Free Boundary Problem via Newton's Method

We develop a Newton-like numerical method to solve a free boundaryproblem of parabolic type. From some numerical tests we concludethat the speed of convergence of this procedure is very high,probably quadratic in the neighbourhood of the solution.