Some Problems On A Class Of Fourth/Sixth Order Nonlinear Diffusion Equations

The higher order nonlinear diffusion equations have a sharp physical background and a rich theoretical connotation. In the last decades, especially in recent twenty years, the study in this direction attracts a large number of mathematicians both in China and abroad.In this thesis, we mainly study the fourth-order Cahn-Hilliard/Allen-Cahn equation, the driven sixth-order Cahn-Hilliard equation with concentration dependent mobility, and the model of oil-water-surfactant mixtures with degenerate mobility.In Chapter 2, we consider a scalar Cahn-Hilliard/Allen-Cahn equation with the boundary condition and the initial value conditionWe discuss the existence of global attractors for problem (14)-(16). The concept about the global attractors is one of the most important ones in dynamical system. The existence of global attractors in a certain sense show the system stability. The main difficulties are causes by the nonlinearity of diffusion part and lower order terms. By using the estimates of semigroups, combining with the iteration technique and the classical existence theorem of global attractors, we prove the existence of global attractors in the Hk (0≤k< 5) space.In Chapter 3, we study the time periodic solutions of the equations in the following where φ(u, t)= The periodic phenomenon exists widely in nature, so the research of the time periodic solutions is important in the theory and application. In this part, we discuss the existence of time periodic solutions of a Cahn-Hilliard/Allen-Cahn equation (17)-(20). Firstly, we introduce operator T. Then we conclude that T is compact and the estimate of the solutions of the equations. Based on the framework of Leray-Schauder fixed point theorem, there is a fixed point in the space C2+a,a/4(Qω. Additionally, we give a suitable upper bound of the L∞ norm of the time periodic solutions.In Chapter 4, we investigate a driven sixth-order Cahn-Hilliard equation with con-centration dependent mobility where and k> 0, v are constants. The equation is supplemented with the boundary and initial value conditions The equation (21) arises naturally as a continuum model for the formation of quantum dots and their faceting [49]. Here u(x, t) denotes the surface slope, and v is proportional to the deposition rate. Our main purpose is to establish the global existence of classical solutions under much general assumptions. The main difficulties for treating the problem are caused by the nonlinearity of the principal part and the lack of maximum principle. The key step is to get a priori estimates on the Holder norm of D2u. Our method is based on uniform Schauder type estimates for local in time solutions via the framework of Campanato spaces.In Chapter 5, we study the model of oil-water-surfactant mixtures with degenerate mobility where > 0, the amphiphile concentration a(u) is approximated by the quadratic function a(u)=γ1 u2, and k> 0,γ1> 0 are constants [20,21]. H’(u)= h(u), The equation is supplemented with the boundary and initial value conditions We study the existence of weak solutions. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After es-tablishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the monotonicity of support of solu-tions are also discussed.