Uniform stability and decay rate of solutions for fractional Cauchy problems

It is known that the solutions of the fractional Cauchy problems on Banach spaces are given by the corresponding fractional resolvent families. To study the asymptotic behaviors of the solutions, we give a sufficient and necessary condition for the uniform stability of fractional resolvent families in terms of the behavior of the resolvents of their generators on Hilbert spaces, which generalizes the classical Gearhart–Prüss theorem. And we completely characterize the exponential growth bound of fractional resolvent families on Hilbert spaces. On Banach spaces, a necessary condition for the uniform stability is also given. Moreover, by using a functional equation for fractional resolvent families, we show a uniformly stable \(\alpha \) $\alpha$ -times resolvent family with generator A will approach to \(\frac{-A^{-1}}{\Gamma (1-\alpha )t^\alpha }\) $\frac{-A^{-1}}{\mathrm{\Gamma }(1-\alpha )t^{\alpha }}$ as \(t \rightarrow +\infty \) $t\to +\infty$ , and the optimal convergence rates are derived.