A Free Boundary Problem With Variable Exponent And Different Moving Parameters

A free boundary problem with variable exponent and and different moving parameters is con-sidered in this paper,and we mainly discuss properties such as local existence and uniqueness of solutions,regularity and long time behavior of solutions.There are two characteristic points in our problem,one is the variable exponent,and the other is that moving parameter may be not equalWe divide the paper into three parts.In chapter 1 we recall some background and progress of research related to our problem.Chapter 2 is the main part of our paper,and consists of three sections,where by making use of classical analysis arguments such as Schauder fixed point theorem and LP theory we establish local existence and uniqueness of solutions,regularity of the solution in §2.1,by employing the comparison principle and upper-sub solution method we deduce some conditions for finite time blowup and global existence in §2.2,and in §2.3 we further investigate the long time behavior of global solutions and give sufficient conditions to ensure the existence of global fast solutions and global slow solutions by applying the comparison principle,upper-sub solution method,scaling argument and energy functions.In chapter 3 we make some conclusions and some expectations about our problem and the related works in the coming.At the same time,we are try to discuss another free boundary problem for localize reaction term with variable exponent and different moving parameters,and what we care about is still local existence and uniqueness of solutions,finite time blowup,global existence and long time behavior of global solutions.