Long time decay analysis of complex-valued fractional abstract evolution equations with delay

Abstract

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.