Some Problems For A Class Of Nonlocal Parabolic Equations

The nonlocal parabolic equations as an important class of integrodif-ferential eq uations,come from a variety of phenomena in nature. In par-ticular,such equations are suggested as mathematical models of physical problems in many fields such as diffusion phenomena in phase transition, epitaxial thin film growing. In the past years,the study in this direction attracts a large number of mathematicians both in China and abroad.By thermodynamic principles,we havem＞0is the mobility or diffusivity. The convective term β.(?)B(u)([6],[19]),In this thesis,we consider some problems on the nonlocal parabolic equationS.we can define the chemical potentialHence,we obtain the nonlocal equationwhere H(u)=∫ΩJ(x-y)dyu(x)-∫ΩJ(x-y)u(y)dy,0≤m(x,t)≤m2,B(u)=|u(x)|q,q>1,T>0,Ω(?)RNis bounded.we do not assume that J is nonnegative but its integral is assumed to be positive. Firstly,we consider the following problemut-β·▽B(u)=div(m(x,t)â–½(H(u)+f(u))), inΩ×(0,T),(1)[m(x,t)â–½(H(U)+f(u))+βB(u)].n=0x∈(?)Ω, t∈(0,T], and the initial value conditionwhich arises naturally as a continuous model for the formation of facets and corners in crystal growth,where H(u)=∫ΩJ(x-y)dyu(X)-∫ΩJ(X y)u(y)dy.We discuss the Neumann boundary problem for a convective nonlo-cal Cahn-Hilliard equation with variable cofficient.The main difficulties with Eq.(1) are that there are nonlocal term,convective term and vari-able cocfficiont. So the maximum principle or comparison thcorom are not available and the equation does not have energy function. In order to overcome these difficulties,we apply Alikakos’iteration method to get some a priori estimates for the solution,then apply the Leray-Schauder fixed point theorem to prove the existence of a solution.We introduce a nonlinear Poincare inequality that should be useful in a variety of settings. By this inequality,we consider the long—term behavior of the solution in the Lp norm.In chapter3,the equation(1)is supplemented by the boundary value conditionsand the initial value condition We first study the nondegenerate problem in Section2,and then establish the existence,uniqueness and continuous dependence on initial data for weak solutions of degenerate problem in Section3. Subsequently,we get an absorbing set in the H1norm in Section4and we prove that there exists a g1obal attractor for the case n=1.In chapter4,we consider m=m(u),β=0,that iswhere m(u)=|u|m,H(u)=∫ΩJ(x-y)dyu(x)-∫ΩJ(x-y)u(y)dy,F represents the(density of)potential energy.Note that the free energy of the equation(2)iswhere J:Rdâ†'R is a smooth function such that J(x)=J(-x).Taking the first variation of X we can define the chemical potential associated with the nonlocal modelwhereThe equation(2)is supplemented by the boundary value conditionsand the initial value condition Since the equation (2) is degenerate when u(x, t)=0, the equation does not admit classical solutions in general. Now, there are no results to nonlocal Cahn-Hilliard equation with degenerate mobility. So, we firstly study the regularization problem. After establishing some necessary uni-form estimates on the approximate solutions, we prove the existence of weak solutions.In chapter5, we study a nonlocal epitaxial thin film growing equationbe nonnegative.Our interest lies in the existence of weak solutions. Because of the degeneracy, we will first consider the non-degenerate problems. As we know, when H(u)=0, the nondegenerate problems have classical solutions, and hence the weak solutions exist. But in the case of problem (3), there are no results to the corresponding non-degenerate problems. Based on the uniform Schauder estimates and using the method of continuity, we obtain the existence of classical solutions for non-degenerate problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.