An application of the theory of scale of Banach spaces

Abstract

he author considers, in an abstract setting, a semilinear evolutionary equation of the form ut+Au=F(u), where A:D(A)⊂X→X is a positive sectorial operator on a Banach space X and F satisfies a suitable local Lipschitz condition. He studies the regularity of solutions when the equation is settled in the scale of Banach spaces Xα=D(Aα). In particular, he focuses on the case when F is a power function. The general results are illustrated by a number of specific applications.