Free Boundary Value Problem For Degenerate Parabolic Equation

The blowup of properties of the solution and the long time behavior of the global solution of the second order parabolic partial differential equations is the most impor-tant part of the theory of the nonlinear partial differential equations. In this article, we will discuss some properties of the solutions of degenerate and singular local parabolic equation with a free boundary, including the local existence and uniqueness of solu-tion, the blow up result of solution and the long time behavior of solution (global fast solution and global slow solution).Firstly, discussing the local existence and uniqueness of solution to a degener-ate and singular local parabolic equation with a free boundary. By establishing the comparison principle and constructing the supersolution and subsolution, in addition, using the contraction mapping theorem and the method of regularization, we get the local solution exists and is unique.Secondly, investigating the blowup property of solution to a degenerate and sin-gular local parabolic equation with a free boundary. By constructing an appropriate lower solution and using the comparison principle and the eigenfunctions method, we can obtain blowup in finite time.Finally, we mainly deal with the existence of global fast solution and global slow solution to a degenerate and singular local parabolic equation with a free boundary. The existence of global fast solution can be established by constructing a suitable upper solution and using the comparison principle and the property of solution to a elliptic boundary value problem, while for the global slow solution, we need to give some a priori estimates for global solutions first.