Local Operators, Space-Time Methods, and Evolution Equations of Diffusion Type

Abstract

This chapter focuses on the local operators, space-time methods, and evolution equations of diffusion types. It considers autonomous and time-dependent evolution equations written with respect to some local operator. When the local operator is locally dissipative and real then the equations are called “evolution equations of diffusion type” (or in short diffusion equations). The general localization results described in the chapter extend earlier results of J. P. Roth proved under rather restrictive conditions and are also of a kind more appropriate for applications to partial differential equations. The chapter discusses earlier basic facts on local operators and the autonomous evolution equations of diffusion type. Some results on propagators and time-dependent evolution equations are presented in the chapter. Results are also discussed in the chapter on localization and on diffusion problems solved for general noncylindrical (x, t)-domains (in particular second order parabolic problems with merely continuous coefficients depending on space and time, posed in very general noncylindrical domains of R N ).