A Sobolev-type Equation, And A High-end Pseudo-parabolic Equation

This article is divided into four chapters:Chapter one succinctly tell us some physical background and what we will do in this article.In chapter two, we discuss the existence of mild and strong solutions of a nonlinear integrodifferential systems of Sobolev type with nonlocal condition in Banach spaces.The equation of Sobolev type is an important type of mathematical physics equations, which has an extensive application. This chapter deals with the existence of mild and strong solutions of a nonlinear integrodifferential systems of Sobolev type with nonlocal condition in Banach spaces. The results are obtained by using semigroup theory and the Schauder fixed point theorem. It gives the definition of resolvent operator, theorems of existence of mild and strong solutions and the proof. These results give some theoretical advice for further controllability of the equation.Chapter three deals with the controllability of a nonlinear integrodifferential systems ofSobolev type with nonlocal condition in Banach spaces. The results are obtained by using semi group theory and the Schauder fixed point theorem.In chapter four, we give an example of a class of inverse problems to determine the coefficient for the non-linear pseudoparabolic equation of high order in two dimensions. Firstly, it is assumed that the coefficient is known, and we work out the general representation of the pseudoparabolic equation given. Then the relationship which unknown coefficient satisfies from additional condition is derived. We deduce the existence and uniqueness of solution of the pseudoparabolic equation at last.