Qualitative Analysis Of Nonlinear Parabolic Partial Differential System Which Is Related To The Subdifferential Operator

In recent decades, nonlinear parabolic partial differential system which is related to the subdifferential operator was an important topic in the field of mathematics. As we know, many models in physics and biology can be described by nonlinear parabolic differential system with subdifferential operator. In particular, phase transition model with constraint and nonlinear parabolic partial differential systems with hysteresis ef-fect have been considered by many scholars. Hysteresis occurs in several phenomena in physics, chemistry, biology, engineering and so on. In physics for instance we encounter it in plasticity, friction, ferromagnetism, ferroelectricity, superconductivity, adsorption and desorption, and in the recently studied materials with shape memory. More gener-ally, hysteresis arises in phase transitions. In engineering application, as the nonlinear parabolic partial differential systems with hysteresis effects reflect the interactions in biological model structure and other physical processes interacting, many mathemati-cians, biologists and physicists studies this issues.By means of subdifferential operator, fixed point theorem and the regularity theory of parabolic systems, this thesis studies the qualitative properties of nonlinear parabolic partial differential system which is related to the subdifferential operator. This thesis is divided into three parts:the well-posed of nonlinear parabolic partial differential system which is related to subdifferential operator are studied in the first part, which is the content of Chapter 2. In Chapter 3, we discuss the periodic solution problem which is related to the subdifferential operator. In the last part, we study the optimal control problem which is also related to subdifferential operator. Specifically, this thesis is organized as follows:In Chapter 1, we introduce some basic theories about subdifferential operator and some models in physics, chemistry and biology. Also in this chapter, the structure and the main results of this thesis are briefly summarized, and we also give some concepts and theoretical results to be used in the thesis.In Chapter 2, we discuss the well-posed of nonlinear parabolic partial differential system which is related to subdifferential operator. The nonlinear parabolic partial differential system contains biological diffusion models with hysteresis effect and p-laplace equation. Firstly, we use the fixed point theorem to discuss the well-posedness of the solutions to a class of biological diffusion models with hysteresis effect We improve the results of [2]. We show the existence of nonnegative solutions to the problem under consideration by using the approximation method when the reaction terms are locally Lipschitz continuous, for most of differential equations of population dynamics do not satisfy the reaction terms are globally Lipschitz continuous.Then, we concern with a class of biological models which consists of nonlinear diffusion equations and a hysteresis operator describing the relationship between some variables of the equations However, since the involving space W01,p（Ω）（p>2） is not a Hilbert space, the theory of [23] can not be applied directly, and thus we show the existence of solutions to system by the limit of some subsequence of solutions （σε,uε） to the suitable approximation problem.In Chapter 3, we discuss the existence of periodic solution to nonlinear parabolic system which is related to subdifferential operator. This Chapter is divided into four parts: The first part, we discuss the periodic solutions of non-isothermal phase separa-tion models with constraint which is related to subdifferential operator It is worth pointing out that in [32, 45, 63, 74], as well as in the most of the literature on the subject, third boundary conditions for temperature 9 or a （9） is essential, the ar-guments therein are not applicable to the case of the homogeneous Neumann boundary condition （[62, 65]） or Dirichlet boundary condition （[46, 91, 92]） and the more delicate analysis is needed. In this paper, we consider the existence of periodic solutions to the above problem by the viscosity and drop the the viscosity term —μ△wt （μ > 0） which plays an important role in obtaining the regularity ω ∈W1,2（0,T; L2（Ω））.In part 2, we concern with the multi-dimensional Cahn-Hilliard equation with a constraint In this paper, we use the subdifferential operator theory other than the qualitative theory of parabolic equation to obtain the existence of periodic solution to multi-dimensional Cahn-Hilliard equation with a constraint. Our results improve the results of Yin et al. （[140]）.In part 3, the existence of periodic solutions is proved by the viscosity approach when the heat force changes periodically in time More precisely, with the help of the subdifferential operator and the Poincare map, the existence of solutions to the approximation problem is shown and the solution of the periodic problem is obtained under consideration by using a passage-to-limit procedure.In part 4, by the viscosity approach, we show the existence of periodic solutions of a class of biological models which consists of a nonlinear diffusion equation and a hysteresis operator. More precisely, we start by analyzing an approximation problem, which is obtained from the above problem by carefully applying a truncation operator to nonlinear terms and adding a diffusion term in the equation for a. Then with the help of the Poincare map, we use Leray-Schauder theorem to prove the existence of a solution （σε,uε） to the approximation problem. Finally, based on the uniform bounds for （σε,uε）, problem admits a solution （σ, u） with u≠0, which is the limit of some subsequence of solutions （σε,Uε）.In Chapter 4, we discuss the optimal control problem which is related to the subdif-ferential operator. This Chapter is divided into four parts: The first part, we introduce the background of optimal control problem.In part 2, we discuss the state-constrained optimal control problem governed by Cahn-Hilliard equations: subject to and F（u）（?） S, where T>0,Ω is a bounded domain in RN（N=1,2,3） with smooth boundary （?）Ω, φ（u）=u3-u,and γ>0 denotes a given parameter.We prove the well-posed for this model with control term by applying the standard methods.We show the relationship between the control problem and its approximation.Moreover,we derive the necessary conditions for the optimal control of our original problem by using the approximate problem.In part 3,we concern with optimal control problem for the phase-field transition system with state constraint and obstacle: subject to F（u）（?） S, where Ω is a bounded domain in in RN（1≤N≤3）with a smooth boundary （?）Ω,φ（u）= 3 — u, 5,7, k > 0 denote given parameters,（?）I[-1,1]（u） is the subdifferential of the indicator function I[₁,1]（u） on the closed interval [—1,1], Bw is a given forcing term on Q, u0,v0 are given initial datums. After showing the relationship between the control problem and its approximation, we derive the pontryagin’s maximum principle for an optimal control of our original problem by using the one of the approximate problems.In part 4, we consider the state constrained optimal control problem for Lengyel-Epstein model with obstacles, subject to and F（u） C S, where Ω is a bounded domain in RN（1 ≤N ≤ 3） with a smooth boundary （?）Ω, φ（u） = u3 — u, δ,γ, k > 0 denote given parameters, （?）I[-1,1（u） is the subdifferential of the indicator function I[-1,1]（u） on the closed interval [-1,1], Bw is a given forcing term on Q,（?）/（?）v is the outward normal derivative on （?）Ω and u0,v0 are given initialov datums. We firstly discuss the well-posed of the above problem and the approximate problems corresponding to the control term. Moreover, we obtain the relationship be-tween the control problem and its approximation. Finally, we derive the pontryagin’s maximum principle for an optimal control of our original problem by using the one of the approximate problems.