Characterizations Of L~p Regularity For Cauchy Problems

Differential equations are widely used in natural sciences and social sciences.The theory of abstract differential equations provides a useful theoretical research framework for a large number of phenomena in practice.It involves the study of non-autonomous systems,quasilinear problems,etc.Regularity is to seek the ap-proximate range of the solution based on some features of non-odd terms,andL~p maximum regularity is used to obtain the best behavior if the non-odd items belong to L~p space.Using maximum regularity,non-autonomous or quasi-semilinear problems can be solved by transforming them into autonomous or linear problems according to the fixed-point theorem and linearization,which is the interest in study-ing maximum regularity.Starting from introducing the basic definitions of L~p maximum regularity for first order Cauchy problems and the corresponding Banach spaces,this paper dis-cusses the existence and uniqueness of the strong solution of first order Cauchy prob-lems with initial values.Then a conclusion is obtained that L~p maximum regularity for non-homogeneous first order Cauchy problems is unrelated to the choice of T and p.Some intermediate proving processes are different from the classical proving methods for first order Cauchy problems,such as the application of the im-plicit function theorem.Subsequently,this paper reviews L~p maximum regularity for second order Cau-chy problems and some important properties,thus discussing the connection and transformation between first order systems and the corresponding second order sys-tems.Then some counterexamples are presented that L~p maximum regularity for first order Cauchy problems is not tenable in particular circumstances.Besides,the appli-cation of L~p maximal regularity for second order Cauchy problems in solving qua-si-semilinear problems is discussed.Finally,this paper introduces the application of L~p regularization in feature selection and probability calculation in SVM.