Bounded imaginary powers of generalized diffusion operators

Abstract

In this paper, we investigate the boundedness of the imaginary powers of four generalized diffusion operators. This key property, which implies the maximal regularity property, allows us to solve both the linear and semilinear Cauchy problems associated with each operator. Our approach relies on semigroup theory, functional calculus, operator sum theory and R-boundedness techniques to establish the boundedness of the imaginary powers of generalized diffusion operators. We then apply the Dore-Venni theorem to solve the linear problem, obtaining a unique solution with maximal regularity. Finally, we tackle the semilinear problem and prove the existence of a unique global solution.