Almost Periodic Type Solutions Of A Class Of Parabolic Equations

The theory of almost periodic functions was first developed by the Danishmathematician H.Bohr, then through S.Bocher, H.Weyl and others’ efforts, thistheory has been greatly developed, and its results are also applied to some areas.Almost periodic functions are first abstracted in the periodic functions. Its type hasdeveloped and experienced the process of asymptotically almost periodic functions,weakly almost periodic functions and pseudo almost periodic functions, almostperiodic type functions are the general term of four function types. Meanwhile, thedimension of variables in the function firstly developed from a dimension tondimension. This article is mainly to study some properties of almost periodic typefunctions under the condition of the variables in the ndimension.The main purpose of this article is to solve such problem whether initial valueof first-class parabolic partial differential equations has almost periodic typesolution, first discuss its existence and uniqueness, and then study its almostperiodic type solutions. The first part main shows almost periodic type functions’historical development and its actual situation. The second part defines the basicconcept of almost periodic type functions, supplies a few basic properties andillustrates some commonly used symbols. The third part firstly proves two importantlemmas, and then uses the lemma for study the initial value problem of linearparabolic partial differential equations with a conclusion that the solution ofequation is almost periodical, which services for following reseach. In part IV, Ifirstly prove the solution of nonlinear parabolic partial differential equation initialvalue exists uniquely, and then from four almost periodic founctions, study whetherinitial value of parabolic partial differential equations has almost periodic solutionfrom four starting almost periodic type solutions of the problem of the study, whichcomes to such an conclusion: For any time T, the equation has a unique almostperiodic solution with variable from one dimension to n dimension.