The existence and regularity theory for abstract semilinear time-fractional evolution equations

In this paper, we investigate the abstract time-fractional evolution equations with Lipschitz perturbations. We prove that the exsictence time of the solution is independent of the Lipschitz constant. We also study the perturbation problems with initial data given by integral formula. This is applied to prove the time differentiability of the solution. The differentiability is essential to use maximum principle when we investigate the quantitative evaluation. Furthermore, we step into the general result for the locally Lipschitz perturbation problems. We guarantee the parallel result to the usual evolution equation theory, the blow up alternative holds. The theory on general Banach spaces enables us to deduce space regularity result easily.