It was widely used in many fields such as Schrodinger equation, heat equation and nonlocal abstract Cauchy problem.

In this paper, we investigate the following non-local Cauchy problem of nonlinear population evolution equation with random periodic migration perturbation,{(p(r,t))/(t)+(p(r,t))/(r)=-μ(r)p(r,t)+f(t,p(r,t)),0<r<rm,t0p(r,0)=p0(r)+g(p(r,t0)),T>t0>0 p(0,t)=β(t)integral from n=r1 to r2 k(r)h(r)p(r,t)dr, t≥0.

n-times integrated C-semigroups and abstract cauchy problem;

We also investigate the relations of n-times integrated C-semigroups and the strong solution of corresponding non-homogeneous abstract Cauchy problem, and the representation of strong solution.

We prove the one-to-one correspondence between mild solutions of a linear partial differential equation with delay and mild solutions of an associated abstract Cauchy problem, and give necessary and sufficient conditions for existence and uniqueness of mild solutions of the linear partial differential equation.

We also give the definitions of solution space and Hille-Yosida space for noncompletely abstract Cauchy problems and show that noncompletely abstract Cauchy problems on its Hille-Yosida space is automatically wellposed, which generalizes deLaubenfels s corresponding results on first ordern- abstract Cauchy problems to second order abstract Cauchy problems.