Robustness of Exponential Dissipation with respect to Small Time Delay

Abstract

We investigate robustness of exponential dissipation for the following general nonlinear evolutionary equation with small time delay: ${\partial }_{t}u+Au=f(u(t),u(t-\tau )).$ We firstly obtain a converse Lyapunov theorem. With the help of it, we establish an important result on robustness of exponential dissipation to small time delay assuming that the nonlinearity is globally Lipschitz.