Well-posedness and stability results for nonlinear abstract evolution equations with time delays

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a general exponential decay estimate for small time delays if the nonlinear term is globally Lipschitz and an exponential decay estimate for solutions starting from small data when the nonlinearity is only locally Lipschitz and the linear part is a negative selfadjoint operator. In the latter case we do not need any restriction on the size of the time delays. In both cases, concrete examples are presented that illustrate our abstract results.