A Free Boundary Problem With Elliptic-parabolic Equations System

This dissertation focuses on the researches on the local classical solution and global classical solution of the problem with a coupled system. We organize it along the following lines:In Chapter 1 we discuss the fixed boundary problem with elliptic equation or elliptic equations (diffraction problem) system. First in Section 1 we obtain the estimate of holder norm in time direction about the system of elliptic equation and elliptic equations by difference method and the result will be used in Chapter 2. Then in Section 2 we discuss the elliptic problem which has time derivative on the boundary condition and obtain the corresponding regularity estimate which is quoted in Chapter 3.In Chapter 2 we deal with the existence and uniqueness of the local classical solution about the multidimensional coupled problem with one parabolic equation and two elliptic equations. Here we mainly use the result in Chapter 1 and through Schauder fixed point principle we prove the result.In Chapter 3 we study the global existence and uniqueness of the multidimensional free boundary problem which include three elliptic equations. Here we also used the result in Chapter I and Schauder fixed point principle . In order to find the global solution we first, find a special solution. Then we make some small perturbation on the initial-boundary condition. And in the end we prove that the problem which has been made some perturbation on the initial-boundary condition also has unique global classical solution.