An Application Of The Theory Of Scale Of Banach Spaces

The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem u_t − Δu = u|u|^s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.