Global existence and blowup of solutions to a free boundary problem for mutualistic model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained. The asymptotic behavior of the free boundary problem is studied. Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.     

Global stability of solutions to a free boundary problem of ductal carcinoma in situ

In the paper we present some remarks on the global stability of steady state solutions to a free boundary model studied by Xu (2004) and also prove some new results of global stability of steady state solutions to the model.     

Global fast and slow solutions of a localized problem with free boundary

In this paper,we consider a localized problem with free boundary for the heat equation in higher space dimensions and heterogeneous environment.For simplicity,we assume that the environment and solution are radially symmetric.First,by using the contraction mapping theorem,we prove that the local solution exists and is unique.Then,some sufficient conditions are given under which the solution will blow up in finite time.Our results indicate that the blowup occurs if the initial data are sufficiently large.Finally,the long time behavior of the global solution is discussed.It is shown that the global fast solution does exist if the initial data are sufficiently small,while the global slow solution is possible if the initial data are suitably large.     

Continuous solutions for a degenerate free boundary problem

We prove existence of continuous solutions for

Global classical solution of Muskat free boundary problem

In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u₁(x,y,t) and u₂(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary

     

Global existence and blowup of a localized problem with free boundary

This paper is concerned with a double fronts free boundary problem for the heat equation with a localized nonlinear reaction term. The local existence and uniqueness of the solution are given by applying the contraction mapping theorem. Then we present some conditions so that the solution blows up in finite time. Finally, the long-time behavior of the global solution is discussed. We show that the solution is global and fast if the initial data is small and that a global slow solution is possible when the initial data is suitably large.     

一个耦合系统自由边界问题的整体古典解

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｄｉｓｃｕｓｓｔｈｅｇｌｏｂａｌｃｌａｓｓｉｃａｌｓｏｌｕｔｉｏｎｏｆａｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｑｕａｓｉｓｔａｔｉｏｎａｒｙｐｒｏｂｌｅｍ .Ｔｈｅｐｒｏｂｌｅｍｃｏｍｅｓｆｒｏｍｔｈｅｄｉｓｃｕｓｓｉｏｎｏｆａｇｒｏｗｔｈｍｏｄｅｌｏｆｓｅｌｆｍａｉｎｔａｉｎｉｎｇｐｒｏｔｏｃｅｌｌ(ｓｅｅ [1— 3])ｉｎｍｕｌｔｉｄｉｍｅｎｓｉｏｎａｌｃａｓｅ .Ｔｈｅｐｒｏｔｏｃｅｌｌｃａｎｂｅｖｉｓｕａｌｉｚｅｄａｓｈａｖｉｎｇａｐｏｒｏｕｓｓｔ…     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.     

Global existence and blowup of a nonlocal problem in space with free boundary

This paper concerns a double fronts free boundary problem for the reaction–diffusion equation with a nonlocal nonlinear reaction term in space. For such a problem, we mainly study the blowup property and global existence of the solutions. Our results show that if the initial value is sufficiently large, then the blowup occurs, while the global fast solution exists for a sufficiently small initial data, and the intermediate case with a suitably large initial data gives the existence of the global slow solution.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

The saddle point property for focusing selfsimilar solutions in a free boundary problem

We establish the saddle point property of the focusing selfsimilar solution of a free boundary problem for the heat equation with free boundary conditions given by and .

     

On the geometric form of solutions of a free boundary problem involving galvanization

In [6], T. I. Vogel studied a free boundary problem originating in the galvanization process. He showed that if the given boundary Γ* is starlike or convex, then so is the free boundary solution Γ. Our purpose is to generalize Vogel's second result by showing (under certain assumptions) that Γ cannot have more (local) maxima or minima (relative to a given direction) than Γ*; also that Γ cannot have more inflection points or greater total curvature than Γ*. The author has already proven analogous results for the Bernoulli free boundary problem in [1], [2] and [3].