Composition and products in vector-valued Sobolev spaces and application

Abstract

In this paper, we will show the properties of composition operators u → f (u) in framework of E-valued Sobolev and Lizorkin–Triebel spaces. Here E is a Banach space. Boundedness and continuity properties will be discussed in a certain detail in Sobolev–Lions type function space concerning two abstract spaces E0 and E in terms of their interpolation. By using these composition properties, we obtain the local and global existence, uniqueness, and Lp-regularity of some nonlinear abstract diffusion equations.