Study Of Several Problems Of Fourth-order Parabolic Equations

In recent years, as the development and innovation of science and technology, the parabolic equation of fourth-order is studied deeply in a lot of subjects and fields of research. Its applications are everywhere and many scholars have paid attention to this model. For example, the thin film equation is derived from the diffusion process of droplet on the solid surface. Cahn-Hilliard equation is used to study the phase transition. Quantum hydrodynamics equation can be applied to the semiconductor quantum fluid mechanics of charge transport.This paper firstly studies a class of fourth-order degenerate parabolic system with Dirichlet boundary: where the boundary value of u is 1. m>0, ε and δ are all positive constants. This is a steady state form for the thin film model with a nonlinear second-order diffusion term. In order to study its existence of solutions, we need to construct a fixed point operator and its definition can be obtained from Lax-Milgram theorem. Moreover, by the compact embedding theorem, the existence of weak solutions will be gained through the Leray-Schauder fixed point theorem. Finally, by choosing appropriate test functions and some important inequalities, the uniqueness is proven.Secondly, we study a relevant fourth-order degenerate parabolic equation The fourth-order term may be greater than 1 and the boundary value of u is l. n,ε,δ, /are all positive constants and m is a nonnegative constant. For the existence, we apply the semi-discrete method. Moreover, when the initial functional is closed to a positive steady-state solution, the uniqueness of solution can be obtained. Finally, the iterative method is used to the semi-discrete problem, and we conclude that the solution is exponentially convergent to a steady solution if the time tends to infinity.Finally, we study a fourth-order degenerate parabolic equation with a nonlinear diffusion: where p>1,m≥0. This equation has appeared in the phase transformation theory and thin film lubrication theory. We also take the semi-discrete method to study its existence. By the existence of the elliptic equations, constructing some approximation solutions, and taking iterative estimates, energy estimation and compactness arguments, the existence and uniqueness of solutions is obtained for the corresponding parabolic equations.