We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $C_0$-semigroup and apply results from Pazy's book (Section 6, Thm. 1.4 and Thm. 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $C_0$-semigroup which depends on time?

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $C_0$-semigroup and apply results from Pazy's book (Section 6, Theorems 1.4 and 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $C_0$-semigroup which depends on time?

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $c_0$-semigroup and apply results from Pazy's book (Section 6, Thm. 1.4 and Thm. 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $c_0$-semigroup which depends on time?

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $C_0$-semigroup and apply results from Pazy's book (Section 6, Thm. 1.4 and Thm. 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $C_0$-semigroup which depends on time?

We have this type of abstract evolution equation $U^{\prime}=A(t)U+F(U)$. If the operator $A$ is independent of time, we can get the local existence by proving that $A$ is infinitesimal generator $c_0$ semi-group and apply two theorems (Th.1.4 and Th.1.5) in Pazy's book section.6. In case $A$ depend of time, can I apply the same theorems? If I can, what is the method to prove that $A$ is infinitesimal generator $c_0$ semi-group which depend of time?

We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $c_0$-semigroup and apply results from Pazy's book (Section 6, Thm. 1.4 and Thm. 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $c_0$-semigroup which depends on time?