Qualitative Properties For Partial Different Equations Problems With Two Kinds Of Boundary Value

The boundary value problems of partial different equation are widely used to inves-tigate many kinds of phenomenon in physics, chemistry, and biology. In this dissertation, we consider the two kinds of boundary value problem of partial different equation. One is the bifurcation of elliptic equations and systems with nonlinear boundary conditions, the other is prey-predator model with freedom boundary.Firstly, we research the bifurcation of elliptic problem with nonlinear boundary con-ditions. By applying the abstract bifurcation theorem, we state the existence of positive solution bifurcating from the trivial solution. By using Rabinowitz’s global bifurcation theorem, we prove the solution curve bifurcation from the trivial solution branch is un-bounded.Secondly, we proposes the bifurcation for two kinds of elliptic system with nonlin-ear boundary conditions. One side, we discuss the bifurcation and stability of elliptic systems with nonlinear boundary conditions. The maximum principle for the systems is established. The principal eigenvalue of elliptic system is positive, the establishmen-t of minor strong maximum principle and the existence of positive strict upper solution are equivalent. The results of equation is expanded to the system. Next, by applying the Crandall-Rabinowitz’s bifurcation theorem, we conclude the existence of smooth so-lutions bifurcating from the trivial solution. Using the sign of the principal eigenvalue for elliptic systems, we research the stability of trivial solution and the stability of bi-furcation solution. On the other hand, we investigative the existence and bifurcation of Lotka-Volterra competition model with nonlinear boundary conditions. By applying the method of upper and lower solutions and the fixed point theory, it prove that if there are the couple upper and lower solutions, then there existence a solution between them. Next, by virtue of the Crandall-Rabinowitz’s bifurcation theorem, we conclude the existence of smooth solutions bifurcating from the semi-trivial solution branch.Finally, we deal with two kinds of prey-predator model with freedom boundary. For the ratio-dependent prey-predator model with freedom boundary conditions, the main ob-jective is to realize the dynamics/variations of the prey, predator and the free boundary. We proof the global existence and regularity of solution. Next, by construction itera-tive function sequence, the long time behaviour of solution is been given. A spreading- vanishing dichotomy holds. For Lotka-Volterra prey-predator model with freedom bound-ary conditions, we considerate the the global existence and regularity of solution. Next, as the time tends to infinite, asymptotic behaviors of solution are discussed. We establish the criteria for spreading and vanishing.