On coupled semilinear evolution systems: Techniques on fractional powers of 4x4 matrices and applications

In this paper, we provide several techniques to explicitly calculate fractional powers of 2x2$2\times 2$ operator matrices Lambda=[(Lambda 11 Lambda 12)] , (Lambda 21 Lambda 22) focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction-diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a 4x4 operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart.