The Global Existence of Nonlinear Evolutionary Equation with Small Delay

On the existence for solutions of evolutionary equations, there are many works and methods 1–7 . For example, the fixed principle 1, 3–5, 7 and Galerkin approximations 2, 6 . They are very classical methods to prove existence and uniqueness. Generally speaking, there are four solution concepts. That is, weak solution, mild solution, strong solution, and classical solution. We can obtain different types for different conditions. For instance 1 , consider the following inhomogeneous initial value problem: