The Global Existence of Nonlinear Evolutionary Equation with Small Delay

Abstract

We investigate the global existence of the delayed nonlinear evolutionary equation ${\partial }_{t}u+Au=f(u(t),u(t-\tau ))$. Our work space is the fractional powers space $X^{\alpha }$. Under the fundamental theorem on sectorial operators, we make use of the fixed-point principle to prove the local existence and uniqueness theorem. Then, the global existence is obtained by Gronwall’s inequality.