Abstract
In this paper, we will show the properties of composition operators \(u\rightarrow f\left(u\right)\) 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 \(E_{0}\) and \(E\) in terms of their interpolation. By using these composition properties, we obtain the local and global existence, uniqueness, and \(L^{p}\) -regularity of some nonlinear abstract diffusion equations.
