Almost Periodic Type Solutions To Parabolic Boundary Value Inverse Problems

The theory of almost periodic functions was first put forward by the Danish mathematician H.Bohr. After several decades of development, the dimension of variable developed from one to n and even n+m(n+m means the former n-dimension of the variable is almost periodic in the equation, and the latter m-dimension is consistent). Almost periodic functions has been also expanded to several types, from almost periodic functions to asymptotically almost periodic functions and then weakly almost periodic functions and eventually developed into pseudo almost periodic functions. This paper will study almost periodic type functions in n+m dimensional space. Almost periodic type functions is a general term for almost periodic functions, asymptotically almost periodic functions, weakly almost periodic functions and pseudo almost periodic functions.The main purpose of this paper is to solve the inverse problem of parabolic partial differential equations under the first boundary value conditions has an almost periodic type solution, and the solution's uniqueness and stability. The first part introduces the development of almost periodic type functions and the origin and practical significance of the inverse problem. The second part gives some basic definitions, theorems and some common symbols in the theory of almost periodic functions. The third part solves that the solution of the parabolic partial differential equations initial value problems is almost periodic type solution. The fourth part solves the almost periodicity to the solution of parabolic partial differential equations with first boundary value, and then study the almost periodicity to the solution of parabolic inverse problem with first boundary value. When we prove the inverse problem, we first prove the situation as time upper bound T is sufficiently small, then extend it to any positive number. Finally, we test the stability of the solution.