Stochastic evolution equation driven by the time-changed Q-Wiener process

In this paper, we investigate a class of stochastic evolution equations driven by the time-changed Q -Wiener process in Hilbert space. Firstly, we establish some new time-changed Gronwall-like inequalities, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. As applications of those inequalities, we prove the existence and uniqueness of the solution to the considered equations under some non-Lipschitz conditions. Moreover, some sufficient conditions ensuring the existence of the global attracting sets and the exponential stability in the mean square of mild solutions for the considered equations are established by introducing approximating systems with strong solutions and using a limiting argument. Finally, some examples are investigated to illustrate the theory.