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.

Blowup, Global Fast and Slow Solutions for a Semilinear Combustible System

In this paper, we investigate a semilinear combustible system ut-duxx = vP, vt-dvxx = uq with double fronts free boundary, where p ≥1,q ≥ 1. For such a prob- lem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq 〉 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p 〉 1, q 〉 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.

Global Well-Posedness of Incompressible Navier-Stokes Equations with Two Slow Variables

In this paper, the global well-posedness of the three-dimensional incompressible Navier-Stokes equations with a linear damping for a class of large initial data slowly varying in two directions are proved by means of a simpler approach.

A Free Boundary Problem of a Semilinear Combustible System with Higher Dimension and Heterogeneous Environment

In this paper, we investigate a free boundary problem of a semilinear combustible system with higher dimension and heterogeneous environment. Such a problem is usually used as a model to describe heat propagation in a two-component combustible mixture In which the free boundary is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup property and the long time behavior of the solution. Our results show that when pq 〉 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p 〉 1, q 〉 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.