Almost Automorphic And Pseudo Almost Automorphic Solutions Of A Class Of Parabolic Partial Differential Equation

The concept of almost periodic function was originated in the 20 s of last century. It’s known as a promotion of periodic function. It was arised by the Danish mathematician H.Bohr in 1924-1926 firstly, and gained great development gradually under the promotion of H.Weyl,S.Bocher, Levitan and other mathematicians generation after generation. The main feature of its development is the expanding of function scope: almost periodic function, consistent almost periodic function,asymptotic almost periodic function, weak almost periodic function pseudo almost periodic function. Every extend in the range of function not only enriches the theory itself, but also promotes its application in practical terms.Almost automorphic function is a kind of new function which is gained when S.Bochner promoted the concept of almost periodic functions naturally. Its development track was accompanied by almost periodic function.Thus, it experienced the stages of almost automorphic function, uniform almost automorphic function, asymptotic almost automorphic function, weak almost automorphic function, pseudo almost automorphic function correspondingly.The theory attract the attention of a large number of mathematical workers and researchers from the beginning of birth.The main purpose of this paper is to investigate the cauchy value problem of a class of nonlinear parabolic partial differential equation whether it has almost automorphic and pseudo almost automorphic solutions. The main works are as follows:Firstly, we extend the traditionly almost automorphic function and pseudo almost automorphic function which is defined in ?®X and ?′X ®X to high dimensional spacen?®X andn m?′?®X, respectively promote its important nature and theorem from one dimensional space to n dimensional space andm +n dimensional space parallelly. Thus we gain the function space which is more widely used.Secondly, we restrict t on a bounded closed subset ofn?.By using the own nature of almost automorphic functions,we obtain the continuity of the limit function g and the uniform convergence of lim()()nnf t t g t®￥+ = and lim()()nng t t f t®￥- = for t. they provides important basis for the following certification.Thirdly, we prove the existence and uniqueness of almost automorphic and pseudo almost automorphic solutions of a class of nonlinearity parabolic partial differential equation.