Composition and Products in Vector-valued Sobolev Spaces and Application

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.