Stability and asymptotics for fractional delay diffusion‐wave equations

Abstract

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both and 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 FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order . However, the study on the long‐time decay rates of the solutions for FDDEs with order is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with , that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, 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 diffusion‐wave equations are . Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results.