在這篇論文中我們考慮下面的抽象柯西問題：
du (t, x)/dt=A (u(t, x)) t >0
u (0, x)=x x屬於C(D(A))
{W(t)}t≧0是Ａ所生成的一個compact C-regularized semigroup作用在一個Banach space X上。若Z ＝ x是這個抽象柯西問題的解空間，設Ｂ是Ａ在Ｚ上的一個有界擾動，我們證明 (A+B)∣z所生成的C-regularized semigroup {S(t)}t≧0在Ｚ上也是compact，並將此結果用在某些偏微分方程上。

In the paper, we consider the abstract Cauchy problem:
du (t, x)/dt=A (u(t, x)) t >0
u (0, x)=x x C(D(A))
where A is an infinitesimal generator of a compact C -regularized semigroup {W(t)}t≧0 on a Banach spaceX. Let Z X be the solution space of the abstract Cauchy problem and let B be a bounded perturbation (on Z) of A. We proved that the C-regularized semigroup {S(t)}t≧0 generated by (A＋B)｜z is also a compact semigroup on Z. We apply this result for certain partial differential equations.