About Initial Boundary Value Problem Of The Mixed Type Partial Differential Equation

In this paper we study the initial boundary value problem of the mixed type equation, which is one of the special study directions in partial differential equation. Much fruitful research work have been made by many scholars at home and abroad on this respect. The problem on changing boundary and non-local problem is an important and interesting issue. This article is divided into five parts to discuss these issues.In part1, we introduce the research background, the present research situation and the research meaning of the mixed type equation.In part2, we state the preliminary knowledge concerning the article.In part3, we discuss the non-local boundary problem on the second-order general form of linear mixed type parabolic equation.In section1, we state the presenting of the problem and correlative suppose.In section2, we discuss the priori estimation of the solution to the problem and the uniqueness and stability of the solution, and obtain theorem 3.2.1, theorem 3.2.2 and theorem3.2.3.In section3, we discuss the existence of the solution, that is, by the basic solution of the parabolic equation to deduce the Volterra integral equation of second type, then by the theory of integral equation group to prove the existence of the solution, obtain theorem 3.3.In part4, we discuss the problem on the changing boundary about the mixed type general form of three-order hyperbolo-parabolic equation with characteristic.In section1, we state the presenting of the problem and the introducing assistant function, that is, the author divided the original problem into three problems by introducing a new unknown function, these are a second-order mixed type hyperbolo-parabolic equation and two first order partial differential equation.In section2, we discuss the uniqueness of the solution. We deduce some kind of estimation for the second-order mixed equation, by the estimation the uniqueness of the solution is proved. The method proposed in this paper is used for first order partial differential equation to prove the uniqueness of the solution. We obtain theorem 4.2.1 and theorem4.2.2.In section3, we discuss the existence of the solution. We deduce the integral form solution for the second-order mixed type equation, by the theory of integral equation group the existence of the solution is proved. The method proposed in this paper is used for first order partial differential equation to prove the existence of the solution. We obtain theorem 4.3.1 and theorem4.3.2.In part5, we discuss the initial boundary value problem of the higher dimensional mixed type parabolo-hyperbolic equation.In setion1, we discuss the priori estimation of the initial boundary value problem of the higher dimensional mixed type parabolo-hyperbolic equation and obtain theorem 5.1.In setion2, we discuss the uniqueness and stability of the initial boundary value problem of the higher dimensional mixed type parabolo-hyperbolic equation and obtain theorem 5.2.1 and theorem 5.2.2.