Periodic Solutions And Maximal Regularity Of Second Order Problems

For the complete second order Cauchy problems, the researches on maximal regu-larity of the problems and existence of the solutions has count for much meaning.In thereal life, various equations including wave equations, beam equations,viscoelasticity equa-tions,strongly damped equations etc. provide abundant background for the complete secondorder equations.Generally to say,partial differential equations often can convert in the in-finite abstract differential equations.Cauchy problem is a kind of abstract form of partialdifferential equations. The maximal regularity of Cauchy problems has been valued since60's in last century, The properties of these problems has become more abundant in the 90's.For the periodic side value problems of Cauchy problems, there has been many conclusionsin 90's in last century and the last few years, also they have been applied extensively.This text mainly uses operator-valued L.Weis multiplier theorem , operator-valuedMarcinkiewicz multiplier theorem and the analysis of sine propagators, also combines to agreat deal of methods, techniques and results of functional analysis in the meantime. Amongthem there are Fourier transformation, R- bounded, UMD space ,Hardy inequality etc..Weadopt some knowledge and methods of maximal regularity of Cauchy problems, combin-ing with the well-known conclusions of second order Cauchy problems, The textual workmainly aims at Lp-maximal regularity, Cθ-regularity, Besov-regularity and wighted space ofthe abstract complete second order Cauchy problems, makes some researches into periodicside value problems , The purpose lies in making it clear the relations of Fourier multipliers,maximal regularity and periodic solutions of the abstract complete second order Cauchyproblems.This text focuses on Lp(0, 2π; X)space, Besov space and weighted Lp-space, discussesthe relations of Fourier multipliers ,maximal regularity and periodic solutions about thecomplete second order Cauchy problems,get the results about multiplier description ofperiodically strong-solution and an equally property of Lp-maximal regularity in weightedspaces. In the real life, these theories lay good foundations for the researches of the posed-ness of the periodic solutions of the quasilinear problems , non- autonomous systems etc.problems, give the thinking of partial existence and the uniqueness of the strong-solution ofthe above-mentioned problems.