Solutions And Regularity For Complete Second Order Equations

Takimg massive classical differential operators and systems as objects, the establishment of maximal regularity theory system to study regularity of their solutions and existence of non-linear problem solutions has extremely vital significance. The abstract differential equation theory has provided the theoryframe for massive concrete phenomena. Each kind of bean equations, including sluggish damping, the structure damping elastic system, has completely provided the rich background for second order equations. This article studies L~P-maximal regularity for completely second order Cauchy problem. Specifically, under L~P-maximal regularity for completely second order Cauchy problem, considering analyticity of the corresponding sine propagator, using the resolvent estimate technology and the fan-shaped expansion to prove that this propagator can be analyticly extended to a fan-shaped region. This has provided a concise essential condition for the establishment of L~P-maximal regularity. Moreover, the connection of L~P-maximal regularity is studied between second order Cauchy problem with initial-value and first system, conveniently used in actual problem, for example bean equations.This article briefly reviewed maximal regularity for first order Cauchy problem, abbreviation regularity. Regularity generally says from three aspects: continual regularity, L~P-regularity (1 < p<∞) and Cθ-regularity ( 0 <θ< 1), C~θ-regularity just being regularity. In order to cause this article to become the integrity of a system, also briefly introduced posedness and other correlation knowledge for second order Cauchy problem. HolderBecause research of maximal regularity for complete second order Cauchy problem only had the distinct progress in recently several years, therefore this article work provide a foundation for further studing completely second order Cauchy problem.