Some Properties Of Reaction Diffusion System With Localized Reaction And Free Boundary

With the development of modern science and technology, there is an increasing number of partial differential equations and systems established in heat conduction, chemical engineering, and combustion theory, which describe physical phenomena in fluid mechanics, quantum mechanics, biochemical kinetics and so on.These kinds of equations (systems) are not all continuously solvable as time increases, because some solutions become unbounded in finite time. In numerical calculations, we also find that blowup occurs frequently. Therefore, it’s of great importance and natural theoretically for us to figure out under what conditions blowup occurs, the global solution exists. Furthermore, when the global solution exists, what’s of great significance here is to make clear the long time behaviors of the solution.Many scholars have studied the blowup problems in a fixed bounded domain in one dimensional case or multiple dimensional case, as well as the corresponding Cauchy problem. Now, we consider the intermediate case between the cases of bounded and unbounded intervals-free boundary problem, which is the core of this paper, and we will give the specific model later. Free boundary problem is always a difficult problem, as the boundary is unknown and is changing as time increases. And we will figure out the free boundary together with the solution. Free boundary problem is different from the growing domain problem, in which the boundary is changing according to a known function.To make this presentation more readable and systematic, we concerns primarily with a nonlinear reaction-diffusion model with a localized reaction term and free boundaries, and the qualitative properties of these models are extensively studied. It consists of three parts.In the first part, we mainly introduce the background and history about the related work.The second part is devoted to a nonlinear reaction-diffusion model with a localized reaction term and free boundaries in one dimension and we mainly give the following results:firstly, the local existence and uniqueness of a classical solution is given. On the second place, we prove that the solution ceases to exist under certain conditions. And our results indicate that blowup occurs if the initial data is sufficiently large.The third part mainly deals with the existence of global fast solution and global slow solution.Our results show that when the initial data is sufficiently small the global fast solution exists, and the global slow solution is possible if the initial data is suitably large.