L^p‐maximal regularity for incomplete second‐order Cauchy problems

The purpose of this paper is to investigate the L^p‐maximal regularity for the abstract incomplete second order problem.

First, the paper gives the definition of the L^p‐maximal regularity for incomplete second‐order Cauchy problems and lists their basic properties based on Chill and Srivastava's recent work for completing second order problem. Second, the paper establishes its characterization by means of Fourier multiplier and the operator‐sum theorem. Finally, it considers an application to quasilinear systems by the regularity and linearization techniques.

Two criteria of L^p‐maximal regularity are obtained, and the existence of the local solution for the second order quasilinear problem is given. In addition, the connection on maximal regularity between second order problems with initial values and that with periodic problems is investigated. A perturbation result is given.

The maximal regularity is an important tool in the theory of non‐linear differential equations. The results obtained in this paper are universal because the operator is not necessarily the generator of a cosine operator function. Using this unifying approach it is possible to clarify the L^p‐maximal regularity and the existence of the solution for some systems described by partial differential equations, such as wave equations.