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We investigate the global existence of the delayed nonlinear evolutionary equation ∂tu+Au=f(u(t),u(t−τ)). Our work space is the fractional powers space Xα. 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

The Global Existence of Nonlinear Evolutionary Equation with Small Delay